Systems and methods for preservation of qubits

ABSTRACT

Embodiments of quantum ring oscillator-based coherence preservation circuits including a cascaded set of stages are described. Embodiments of such quantum ring oscillator-based coherence preservation circuits allow the internal (superpositioned) quantum state information of stored qubits to be preserved over long periods of time and present options for the measurement and potential correction of both deterministic and non-deterministic errors without disturbing the quantum information stored in the structure itself.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of, and claims a benefit of priorityunder 35 U.S.C. 120 of the filing date of U.S. patent application Ser.No. 16/748,481, filed Jan. 21, 2020, entitled “SYSTEMS AND METHODS FORPRESERVATION OF QUBITS”, issued as U.S. Pat. No. 10,878,333, which is acontinuation of, and claims a benefit of priority under 35 U.S.C. 120 ofthe filing date of U.S. patent application Ser. No. 15/965,286, filedApr. 27, 2018, entitled “SYSTEMS AND METHODS FOR PRESERVATION OFQUBITS”, issued as U.S. Pat. No. 10,579,936, which claims a benefit ofpriority under 35 U.S.C. § 119 to U.S. Provisional Patent ApplicationNo. 62/491,815 filed Apr. 28, 2017, entitled “QUANTUM STATE OSCILLATORSAND METHODS FOR OPERATION AND CONSTRUCTION OF SAME”, the entire contentsof which are hereby expressly incorporated by reference for allpurposes.

TECHNICAL FIELD

This disclosure relates generally to quantum computing. In particular,this disclosure relates to embodiments of systems and methods forpreserving quantum coherence of a qubit.

BACKGROUND

Certain computational problems, such as the factoring of large numbers,cannot easily be solved using conventional computers due to the timerequired to complete the computation. It has, however, been shown thatquantum computers can use non-classical algorithmic methods to provideefficient solutions to certain of these types of computational problems.

The fundamental unit of quantum information in a quantum computer iscalled a quantum bit, or qubit. Quantum computers can use a binaryrepresentation of numbers, just as conventional binary computers. Inaddition, quantum systems can also make us of use multi-valued logic anddata, in which case, the atomic quantum datum is referred to as a“qudit”. An individual qubit or qudit datum can be physicallyrepresented by the state of a quantum system. However, in a quantumsystem, the datum can be considered to be in more than one of thepossible states at any single given time. Thus, in the case of a qubit,the datum can be in a state that represents both a zero and a one at thesame time. This state is referred to as superposition. Quantumsuperpositions of this kind are fundamentally different from classicaldata representations, even when classical probabilities are taken intoaccount. It is only when a quantum datum is observed that its value“collapses” into a well-defined, single state. This “collapse” isreferred to as decoherence.

Thus, while bits in the classical computing model always have awell-defined value (e.g., 0 or 1), qubits in superposition have somesimultaneous probability of being in both of the two states representing0 and 1. It is customary to represent the general state of a quantumsystem by |ψ>, and let |0>and |1>represent the quantum statescorresponding to the values 0 and 1, respectively. Quantum mechanicsallows superpositions of these two states, given by|ψ>=α|0>+β├where α and β are complex numbers. In this case, the probability ofobserving the system in the state |0>is equal to α2 the probability ofthe state |1>is β².

Quantum computers may utilize physical particles to represent orimplement these qubits or qudits. One example is the spin of anelectron, wherein the up or down spin can correspond to a 0, a 1, or asuperposition of states in which it is both up and down at the sametime. Performing a calculation using the electron may essentiallyperform the operation simultaneously for both a 0 and a 1. Similarly, inthe photonic approach to quantum computing, a “0” may be represented bythe possibility of observing a single photon in a given path, whereasthe potential for observing the same photon in a different path mayrepresent a “1”.

For example, consider a single photon passing through an interferometerwith two paths, with phase shifts φ₁ and φ₂ inserted in the two pathsrespectively. A beam splitter gives a 50% probability that the photonwill travel in one path or the other. If a measurement is made todetermine where the photon is located, it will be found in only one ofthe two paths. But if no such measurement is made, a single photon cansomehow experience both phase shifts φ₁ and φ₂ simultaneously. Thissuggests that in some sense a photon must be located in both pathssimultaneously if no measurement is made to determine its position. Thiseffect can be experimentally verified by observing the interferencepattern resulting from the interaction of the two paths when only asingle photon is allowed to transit through the apparatus at a giventime. Of course, if there are more than a single pair of possiblephotonic paths, then the resulting system can be said to represent aqudit.

One of the most challenging problems with practical quantum computing,however, is the realization of the physical system that represents thequbits themselves. More specifically, the scale at which qubits aretypically implemented (e.g., a single electron, a single photon, etc.)means that any perturbations in the qubit caused by unwantedinteractions with the environment (e.g., temperature, magnetic field,etc.) may result in an alteration to the state of the qubit or evendecoherence. Quantum coherence preservation (e.g., maintenance orstorage of the qubit in a quantum state for any useful time period)within a single qubit (or multiple qubits) is thus a major obstacle tothe useful implementation of quantum computing. Exacerbating the problemis the fact that when several such qubits are placed in close proximityto one another they can potentially mutually interfere (e.g.,electromagnetically) with each other and, thereby, affect adjacentqubits. In some cases, that interference is desired (in the case ofquantum data computation operations, for example), but in the case wherethat interference is uncontrolled, then it can lead to incorrectcomputational results. Such unwanted interference effects are sometimesreferred to as quantum gate or processing infidelities.

Accordingly, there is a need to for systems and methods that can bothpreserve coherence of a qubit from external interference as well as toallow the operations on that qubit to be corrected in the presence ofunwanted quantum operational infidelities.

SUMMARY

To address this need, among others, attention is directed to embodimentsof systems and methods for preserving quantum coherence as depictedherein. A bit of additional context may be useful to an understanding ofsuch embodiments. The generation and subsequent use of superimposedquantum states (qubits) in quantum circuits has been studied for severaldecades. Moreover, as discussed, there are many uses for such quantumcircuits; some of which exhibit significant advantages over traditional(classical) circuits. However, maintaining the superpositioncharacteristic of such qubits over useful amounts of time in practicalenvironments has been a source of difficulty. One problem is thatinteractions of the superimposed qubits with other systems can result inan unintentional “measurement” resulting in loss of superposition. Othererrors, both deterministic and non-deterministic, may result in thedecoherence of qubits or other errors related to their storage. Forexample, on the deterministic side, gate fidelity issues within thegates of a quantum circuit may result in unwanted “corruption” of adesired quantum operation on a set of qubits. Also, quantum circuits maybe effected by other circuits in a system (e.g., through quantum leakageor the like) which could shift or alter neighboring qubits stored inthese circuits. The isolation of the qubit itself from external orenvironmental influence can be quite problematic and may require extrememeasures, such as substantial magnetic shielding or sub-absolute-zerocryogenic apparatus, where the possibility of external magnetic fieldsor infrared radiation from the apparatus itself interacting with thestored qubit is thus minimized.

Thus, it is desirable to create a system or quantum circuits where thequantum state can be maintained for as long as necessary in order tomake use of the qubits in subsequent quantum computations or otherapplications. In other words, it is desirable to have quantum circuitswith a longer longitudinal coherence time (e.g., T₁ time) and transversecoherence time (e.g., T₂ time). In simple terms, the T₁ time can beconsidered the “native” decoherence (or “relaxation”) time of the qubitinformation carrier. In the same vein, the T₂ time can be considered tobe the overall decoherence time (which also includes the effect ofexternal influences on the qubit carrier). Thus, the difference betweenthe T₁ and T₂ times can be considered as a relative measure of thesystem's isolation from the external environment. Other measurements(such as the T₂* time) may include the effect of neighboring-qubitinterference.

Embodiments of the systems, structures and quantum circuits disclosedherein achieve longer T₁ or T₂ times among other advantages through thecontinuous regeneration of a particular quantum state by repeatedlyevolving the qubit carriers to the desired state and subsequently backto an eigenstate or basis state. In particular, embodiments may comprisea set of cascaded stages with the output of the last linear stage of thecircuit being fed back, or provided as input, to the first linear stagein the set of cascaded stages. Because the subsequent states produced bythe individual stages differ in a repeatable pattern, a quantumoscillator results wherein the final basis state may be affected by thesuperimposed state that is desired to be retained in the interior of theapparatus. Since the basis state value may be known (and thus, may bemeasured without disturbing the superpositioned states), its state maynonetheless also be affected by gate infidelities inherent in thecascade of quantum processing stages required to evolve the qubit backto the basis state. Thus, if the qubit carrier does not fully return toan eigenstate, due to gate fidelity errors in the cascaded circuit, thatfact may be measured and appropriate corrective action applied to thecircuit. Also, the act of measurement of the basis state output of thecascaded circuit may itself contribute to the correction of thepotential gate fidelity errors, which is an application of the so-called“Quantum Zeno Effect”. It should be noted that, while the Quantum ZenoEffect may be responsible for correction of some small gate infidelityerrors, if the errors are substantial, then they will overwhelm theeffect itself and then must be corrected via active measures.

Accordingly, embodiments as disclosed herein provide quantum circuitsthat repeatedly evolve oscillating qubit basis states over time forsingle and multiple qubits. These quantum circuits are characterized bycontinuously regenerating a quantum basis state that oscillates among asubset of different basis states while also evolving any superimposedand entangled states at other points in the circuit. This recirculationallows such quantum preservation circuits to operate in a closed-loopfashion. This permits the application of feedback as well as feedforwardanalysis and control theory techniques for real-time improvements inoperational optimization and stability of the circuit. These quantumcoherence preservation circuits or quantum ring oscillators (which willbe understood to be interchangeable terms as used herein) may be widelyapplicable for various functions in a quantum computing system orotherwise. For the purposes of simplicity, embodiments have beendescribed herein with respect to qubits, but it will be noted that thesame principles and embodiments described herein can nonetheless be inorder to represent qudits (multi-valued quantum data) as well as qubits(binary-valued quantum data).

In one embodiment, a quantum ring oscillator circuit is provided forquantum coherence preservation of single qubit. Specifically, a NOT gateor Pauli-X gate acts on a single qubit. It is the quantum equivalent ofthe NOT gate for classical computers (with respect to the standard basis|0

, |1

. It equates to a rotation of the Bloch sphere around the X-axis by πradians. It maps |0

to |1

and |1

to |0

. It is represented by the Pauli matrix:

$X = {\begin{bmatrix}0 & 1 \\1 & 0\end{bmatrix}.}$A square root of NOT gate acts on a single qubit and is represented by aunitary matrix that, multiplied by itself, yields X of the NOT gate:

$\sqrt{X} = {\sqrt{NOT} = {{\frac{1}{2}\begin{bmatrix}{1 + i} & {1 - i} \\{1 - i} & {1 + i}\end{bmatrix}}.}}$

Accordingly, in one embodiment a quantum coherence preservation circuitthat is a single bit oscillator may include two cascaded linear stages,each linear stage comprised of a square root of NOT gate. The two squareroot of NOT gates that are cascaded with the output of the last squareroot of NOT gate provided to (or fed back to) the input of the firstsquare root of NOT gate. Specifically, in these embodiments, the outputof a first square root of NOT gate may be coupled to a second squareroot of NOT gate. The output of the second square root of NOT gate iscoupled to the input of the first square root of NOT gate. Thus, if theinput qubit to the first square root of NOT gate is in a first state,the output of the first square root of NOT gate is a qubit in a secondstate provided to the second square root of NOT gate, the qubit that isoutput of the second square root of NOT gate is in a third state, wherethe third state is the opposite (e.g., NOT) of the first state. Thequbit in the third state is then fed back on the output of the secondsquare root of NOT gate to the input of the first square root of NOTgate. After the second pass through this quantum circuit, the qubit willbe in the first state again. In other words, the qubit has gone throughthe equivalent of two NOT gates. In this manner, the state of the qubitis oscillated between the first state and its opposite (e.g., NOT thefirst state) and can be maintained in the quantum circuit whilepreserving the first state of the qubit maintained therein (e.g., thestate in which the qubit is initially input to the circuit).

In another embodiment, a quantum coherence preservation circuit that ismade up of a single qubit oscillator may include cascaded stages, eachstage comprising a Hadamard gate. The two Hadamard gates are cascadedwith the output of the last Hadamard gate provided to (or fed back to)the input of the first Hadamard gate. Specifically, in theseembodiments, the output of a first Hadamard gate may be coupled to asecond Hadamard gate. The output of the second Hadamard gate is coupledto the input of the first Hadamard gate. Thus, if the input qubit to thefirst Hadamard gate is in a first state, the output of the firstHadamard is a qubit in a second state provided to the second Hadamardgate, the qubit that is output of the second Hadamard gate is in a thirdstate, where the third state is equivalent to the first state, since theHadamard operation is its own inverse. The qubit in the third stateequivalent to the first state is then fed back on the output of thesecond Hadamard gate to the input of the first Hadamard gate. In thismanner, the state of the qubit is oscillated between the first andsecond states (which are equivalent) and can be maintained in thequantum circuit while preserving the first state of the qubit maintainedtherein (e.g., the state in which the qubit is initially input to thecircuit).

Embodiments of quantum circuits that preserve coherence of two or morequbits through the oscillation of states are also disclosed. Certain ofthese embodiments may feed the output of one or more gates of thequantum circuit to one or more inputs of one or more quantum gates ofthe quantum circuit. In particular, embodiments as described may utilizea quantum circuit that produces linear combinations of Bell States asoutput values. Various embodiments of this circuit may involvecontinuous regeneration or circulation of qubits that undergo successivesuperposition, entanglement and then decoherence operations. Theregenerative nature of this circuit and the recirculation allows thecircuit to operate in a closed-loop fashion. This permits theapplication of feedback as well as feedforward analysis and controltheory techniques for real-time improvements in operational optimizationand stability of the circuit. Because of its structure (a cascaded setof Bell-State generators) and due to its alternating basis stateoutputs, the dual qubit embodiment of this kind of regenerative quantumcircuit is referred to as a “Bell State Oscillator” (BSO).

Certain embodiments of a BSO as disclosed can be used to generate andpreserve a pair of entangled qubits, and thus may be thought of as aqubit storage device or cell that holds a pair of entangled qubits. Morespecifically, some embodiments of a BSO may continuously generate (orregenerate) and circulate pairs of qubits in a feedback loop. Such a BSOmay, for example, include a set of cascaded Bell State generatorcircuits, with each Bell State generator circuit providing the input tothe subsequent Bell State generator circuit, and the output of the finalBell State generator in the chain (which will have evolved back to abasis state) coupled back to the input of the first Bell State generatorcircuit in the chain.

In one embodiment, a quantum circuit for a dual qubit oscillator mayinclude a BSO having four cascaded linear stages, each linear stagecomprising a Bell State generator. More particularly, in one embodiment,a BSO includes a first Bell State generator, comprising a first Hadamardgate and a first CNOT gate, the first Hadamard gate having an input andan output and the first CNOT gate having an input and an output. The BSOalso includes a second Bell State generator, comprising a secondHadamard gate and a second CNOT gate, the second Hadamard gate having aninput and an output and the second CNOT gate having an input and anoutput, wherein the input of the second Hadamard gate is coupled to theoutput of the first Hadamard gate of the first Bell State generator andthe input of the second CNOT gate is coupled to the output of the firstCNOT gate of the first Bell State generator. The BSO may also include athird Bell State generator, comprising a third Hadamard gate and a thirdCNOT gate, the third Hadamard gate having an input and an output and thethird CNOT gate having an input and an output, wherein the input of thethird Hadamard gate is coupled to the output of the second Hadamard gateof the second Bell State generator and the input of the third CNOT gateis coupled to the output of the second CNOT gate of the second BellState generator. The BSO may further include a fourth Bell Stategenerator, comprising a fourth Hadamard gate and a fourth CNOT gate, thefourth Hadamard gate having an input and an output and the fourth CNOTgate having an input and an output, wherein the input of the fourthHadamard gate is coupled to the output of the third Hadamard gate of thethird Bell State generator and the input of the fourth CNOT gate iscoupled to the output of the third CNOT gate of the third Bell Stategenerator, and wherein the input of the first Hadamard gate of the firstBell State generator is coupled to the output of the fourth Hadamardgate of the fourth Bell State generator and the input of the first CNOTgate of the first Bell State generator is coupled to the output of thefourth CNOT gate of the fourth Bell State generator.

Other embodiments of quantum circuits herein may utilize Greenberger,Home and Zeilinger (GHZ) state generators. The GHZ state generator caneffectively be considered as a 3-qubit version of the Bell Stategenerator, the output of which is a maximally-entangled qubit triplet(as opposed to the Bell State generator output, which is a maximallyentangled qubit pair). One implementation of a GHZ state generator mayinclude a Hadamard gate having an input and an output and a first CNOTgate having an input and an output and controlled by the output of theHadamard gate and a second CNOT gate having an input and an output andcontrolled by the output of the Hadamard gate. Thus, an embodiment of aquantum oscillator for three qubits may include a cascaded set ofstages, each stage comprising a GHZ stage generator, and may be referredto as a GHZ state oscillator or GSO. Certain embodiments of a GSO asdisclosed can be used to generate and preserve three maximally-entangledqubits, and thus may be thought of as a qubit storage device or cellthat holds the three entangled qubits, with similar properties as boththe single-qubit and dual-qubit oscillator circuits described earlier.More specifically, some embodiments of a GSO may continuously generate(or regenerate) and circulate three qubits in a feedback loop. Such aGSO may, for example, include a set of cascaded GHZ state generators,with each GHZ state generator circuit providing the input to thesubsequent GHZ state generator circuit, and the output of the final GHZstate generator in the chain (which will be evolved back to a basisstate) coupled back to the input of the first GHZ state generatorcircuit in the chain.

Specifically, in one embodiment, a GSO may include a first GHZ stategenerator having a Hadamard gate with an input and an output, a firstCNOT gate having an input and an output, the control of the first CNOTgate coupled to the output of the Hadamard gate, and a second first CNOTgate having an input and an output, the control of the second CNOT gatecoupled to the output of the Hadamard gate. A second GHZ state generatorhaving is cascaded with the first GHZ state generator. The second GHZstate generator includes a Hadamard gate with an input and an output, afirst CNOT gate having an input and an output, the control of the firstCNOT gate coupled to the output of the Hadamard gate, and a second CNOTgate having an input and an output, the control of the second CNOT gatecoupled to the output of the Hadamard gate. The second GHZ stategenerator is cascaded with the first GHZ state generator by coupling theoutput of the first CNOT gate of the first GHZ state generator to theinput of the first CNOT gate of the second GHZ state generator, theoutput of the second CNOT gate of the first GHZ state generator to theinput of the second CNOT gate of the second GHZ state generator and theoutput of the Hadamard gate of the first GHZ state generator to theinput of the Hadamard gate of the second GHZ state. A third GHZ stategenerator is cascaded with the second GHZ state generator and a fourthGHZ state generator is cascaded with the third GHZ state generator in asimilar manner. A feedback path of the GSO couples the output of thefourth GHZ state generator to the input of the first GHZ state generatorby coupling the output of the first CNOT gate of the fourth GHZ stategenerator to the input of the first CNOT gate of the first GHZ stategenerator, the output of the second CNOT gate of the fourth GHZ stategenerator to the input of the second CNOT gate of the first GHZ stategenerator and the output of the Hadamard gate of the fourth GHZ stategenerator to the input of the Hadamard gate of the first GHZ stategenerator.

In a similar manner, it can be seen that the BSO and GSO structuresdescribed earlier can themselves be extended to incorporate furtherqubits. Thus, if we replace each of the four GHZ State generator stagesof the GSO described earlier with their four-qubit counterparts, thenthe resulting circuit will produce similar results. Specifically, theBloch sphere rotations of the input qubits will result in a basis stateoutput at the conclusion of the fourth stage in the chain, much like theBSO and GSO circuits. Thus, in order to create larger-sized groups ofmaximally-entangled qubits, the length of the cascaded chain does notincrease.

This overall circuit architecture of four-stage chains ofmaximally-entangled state generators can thus be seen to represent amethod for creating and maintaining larger-sized entangled qubit“words”, where the size of the circuit that creates themaximally-entangled qubits grows linearly with the size of the desiredentangled qubit word. This linear scaling is in contrast to thequadratic or even exponential growth of circuits that are typicallyrequired to produce larger-sized maximally-entangled qubit words. Thesecircuits containing various numbers of 4-stage chains of cascadedmaximally-entangled state generators may be referred to a quantum ringoscillators, due to their similarity to classical binary ring oscillatorcircuits.

Because embodiments of the quantum coherence preservation circuits asdiscussed operate by oscillating the quantum qubits through a definedset of states, certain types of errors that may decrease quantumcoherence time may be effectively measured and then corrected withoutcausing decoherence of the qubit. More specifically, in theseembodiments, when a qubit is input to the quantum circuit in a basisstate, the input received on the feedback loop will be a defined basisstate. Thus, for example, in a quantum circuit for coherencepreservation of single qubit using two cascaded square root of NOTgates, if a qubit is input in a basis state (e.g., |0

or |1

) the output of the second square root of NOT gate that is fed back tothe input of the first square root of NOT gate will be in an oppositebasis state (e.g., |1

or |0

). As another example, in a quantum circuit for coherence preservationof single qubit using two cascaded Hadamard gates, if a qubit is inputin a basis state (e.g., |0

or |1

) the output of the second Hadamard gate that is fed back to the inputof the first square root of NOT gate will be in the same basis state(e.g., |0

or |1

).

Accordingly, by injecting one or more qubits or sequences of qubits inbasis states into such a quantum ring oscillator circuit and measuringthe state of these qubits on the feedback path of the circuit,deterministic errors in the circuit may be determined and potentiallycorrected. Specifically, by injecting one or more qubits in a knownbasis state (which may be referred to as error detection qubits),expected values (e.g., the same or opposite basis state) for those errordetection qubits on the feedback path may be determined. Any error(e.g., deviations from the expected value) in these error detectionqubits as measured on the feedback path of the quantum ring oscillatorcircuit are usually due to gate fidelity issues or other deterministicerrors. Based on any measured errors determined from the differencebetween the expected basis state for these error detection qubits andthe measured states of the error correction qubits, errors in thequantum ring oscillator circuit (such as phase shift errors or the like)may be determined. Error correction circuitry included in the quantumring oscillator circuit may apply error correction to the qubit carriersignal path to correct for these measured errors. For example, adeterministic phase shift may be applied to the qubit carrier signalpath to correct for these measured errors. Other techniques for errorcorrection in a quantum circuit are known in the art and are fullycontemplated herein.

The qubit state is thus cycled back and forth between the initial stateand its inverse state. Notably, if the initial state is one where thequbit carrier is in superposition, then the inverse state is similarlyin superposition, whereas the intermediate state may be in (orapproximately in) a basis or eigenstate. In this manner, if a singlequbit in superposition is immediately preceded by and immediatelyfollowed by a carrier that is in a basis state, then any gateinfidelities inflicted on both of the basis state carriers will mostlikely also affect the carrier in superposition.

Thus, because of the feedback path included in embodiments of thequantum ring oscillator circuits as disclosed herein, we can use themeasured gate infidelities of the basis-state carriers to estimate theerrors imposed on the qubit in superposition without either having tomeasure the qubit (which would cause decoherence) or to interrupt thenormal circuit operation and initiate a system calibration cycle.

In other, words, the error correction may be applied by the errorcorrection circuitry in a continuous manner while the quantum circuit isin an operational state (e.g., being used for quantum computation)without disturbing the quantum information being used by the quantumsystem or taking the quantum circuit or quantum system offline.Moreover, this error determination and correction may take place atcertain regular intervals or substantially continuously as the quantumcircuit is operating. Thus, as the errors in the quantum ring oscillatorcircuit change (e.g., as the circuit heats up, the qubit carrier pathmay lengthen or the coupling between the circuit and its operatingenvironment may change, etc.) the differing or changing errors may beaccounted for, and the error correction adjusted. In this manner, it ispossible to run real time adjustment of quantum storage circuits toaccount for deterministic errors and increase the length of time ofquantum coherence preservation.

According to some embodiments then, error measurement circuitry mayinject or place one or more qubits or patterns of error detection qubitsin a known basis state before or after a qubit which is intended to bestored or otherwise maintained in the quantum ring oscillator circuit.This qubit may be referred to as an operational or stored qubit and maybe, for example, a qubit used in the performance of generalized quantumcomputing operations by the quantum circuit or a system that includesthe quantum circuit. Any “auxiliary” or “ancilla” error detection qubitsmay be injected into the feedback loop of the quantum ring oscillatorcircuit such that they are initially provided in a basis state at theinput of the first gate or other quantum structure of the circuit.

In another embodiment, however, in order to better isolate any gateinfidelities (between the different stages of the cascaded circuit), theancilla qubits may potentially be introduced into the feedback system inone or more Bell States; thus causing the circuit to produce basis stateintermediate values at any point in the cascade (which can thus bemeasured at that intermediate point in the interior of the cascade). Theerror measurement function of the quantum ring oscillator circuit maythen determine the expected values (e.g., the same or opposite basisstate) for those error detection qubits not only in the feedback path,but at any point along the cascade. The state of these error detectionqubits may then be measured and a difference between the measured errordetection qubits and the expected value for those error detection qubitsdetermined. Based on this difference, errors in the quantum ringoscillator circuit (such as phase shift errors or the like) may bedetermined. Error correction circuitry included in the quantum ringoscillator may apply error correction to the qubit carrier signal pathto correct for these deterministic errors. For example, a deterministicphase shift may be applied to the qubit carrier signal path to correctfor these measured errors.

As it is the error detection qubits that are measured, thesemeasurements and corrections can be made without decohereing orotherwise affecting any operational or stored qubits in the quantum ringoscillator circuit. Since this error correction process isdeterministic, then it may be implemented in classical circuitry. Suchclassical control process is not only simpler to implement thanquantum-based processing, it also allows us to take advantage ofclassical control theory mathematical techniques that have beenwell-studied and continually improved over the better part of the lastcentury. This control function may also be implemented inelectronics-based systems, rather than having to be implemented in the“native” qubit carrier mechanism (.e.g. in photonics). In this way, theerror correction applied by classical error correction control circuitryto adjust the quantum ring oscillator serves to counteract the issuesthat may be caused by the circuit's gate infidelities and thus, toincrease the overall coherence time of any such operational or storedqubits in the circuit.

Additionally, such error correction may be performed repeatedly, orsubstantially continuously, utilizing the same error detection qubitsafter they have passed through the circuit one or more times. Thismulti-pass error correction mechanism will allow for the correction ofsmaller gate infidelity errors than may be normally detectable, sincethe errors would be cumulative. As the initial basis state of theseerror detection qubits is known, the expected values for these errordetection qubits at any given point (e.g., after a given number ofpasses through the closed-loop circuit) may be measured and used toestimate the deterministic errors in the quantum ring oscillatorcircuit. The control circuitry can then apply higher-precision errorcorrection to adjust for this longer-term cumulative determined error.As before, such error correction can be applied and adjusted inreal-time to the qubit carrier signal path while the quantum circuit isin an on-line or operational state without decohereing or otherwiseaffecting any operational or stored qubits in the circuit, serving togreatly increase the coherence time of those operational or storedqubits.

While deterministic errors may be accounted for according to certainembodiments, it may be difficult to account for non-deterministicerrors. For example, in quantum circuits implemented using photonicinformation carriers, one non-deterministic error is photonic loss. Overtime, the chance or probability that a photon will interact with anyatom in the overall data path and cause the qubit to decohere increases.However, it is impossible in a quantum circuit to utilize repeaters asare typically used with in fibre channel or the like, as the use of suchrepeaters would cause the qubit to decohere. Moreover, the no-cloningtheorem states that it is impossible to create an identical copy of anarbitrary unknown quantum state.

Embodiments as disclosed herein may help to determine the statisticallikelihood of the possibility of such non-deterministic errors,including the particular issue of photonic loss. By implementing suchonline error measurements, the operational characteristics of thecircuit may be analyzed in real time to help determine the statisticalprobability of photonic loss while the circuit is in operation. In thecase where the average lifetime of a photonic carrier in the circuit isknown in real time, then further optimizations to the circuit maypotentially be applied that can help to lengthen the average timebetween photonic loss events. This kind of optimization may furtherincrease the coherence time of stored qubits as well as givingapproximate measurements of the probability that a given photoniccarrier has not been absorbed, even though this measurement wouldclearly not give a definite indication whether or not a particularphoton has been absorbed or scattered. However, even if the measurementonly give a range of probabilities of photonic loss, this is nonethelesshighly useful information.

In the case where the photonic qubit carrier is used to represent anexternally-supplied qubit, we transfer that external qubit to the localcircuit by initiating a controlled swap of quantum states between onephoton carrying an external qubit and another (locally-sourced) photon.By swapping the state of the external qubit to a different photon (e.g.,a local photon), we can detect that the local photon's state may havechanged if it is entangled with other local photons. If this isaccomplished correctly (i.e., using qubit clusters), then we can beassured that the external qubit information has been transferred to thelocal circuit without causing decoherence of the external qubit. Thisquantum state swap may be accomplished using a number of mechanisms.

In one embodiment, for example, a Fredkin gate may be used to couple twoquantum ring oscillator circuits, a process which we will refer to as“qubit injection”. A qubit may be injected onto a first quantum ringoscillator circuit and the success of the qubit transfer confirmed.Then, after a number of cycles of this qubit around the first of thequantum oscillator circuit, the qubit may then be transferred to adifferent quantum ring oscillator, where the success of the transfer maythen again be determined. In this way, although we may not be able toprevent loss of the qubit information due to photonic loss, we maynonetheless be able to determine whether or not the information hasactually been lost without measuring it.

Thus, embodiment of the quantum circuits as disclosed herein providesystems a for evolving the quantum states of one or more qubits in adeterministic and repeating fashion such that the time interval requiredfor them to maintain coherence is minimized, thus decreasing thelikelihood that decoherence or unintentional observations occur.Additionally, such quantum circuits may provide the capability tomeasure or otherwise utilize qubits that oscillate among basis stateswithout disturbing the coherency of the quantum state in other portionsof the structure and to provide a convenient means for the injection andextraction of the quantum information carriers without disturbing ordestroying the functionality of the system or any system thatincorporates such quantum circuits.

Moreover, embodiments of these quantum circuits for the preservation ofcoherence of one or more qubits may be easily utilized as supportingsubcircuits in other systems. In addition to the application of theiruse as, for example, a synchronization circuit, additional operators (ortheir inverses) may be added in the feedforward portion of the structureto cause intentional and arbitrary quantum states to be continuouslyregenerated. Such a quantum state storage capability is significantsince quantum storage is a fundamental requirement in many quantuminformation processing designs. The overall state of the oscillators canbe observed by measuring the extracted qubit without affecting otherinternal quantum states.

In one embodiment, a system for the quantum coherence preservation of aqubit, can include a quantum oscillator including a plurality ofcascaded stages, each stage including a quantum circuit having an inputand an output and adapted to evolve a qubit between a first state on theinput and a second state on the output wherein the stages are cascadedsuch that the input of one stage is coupled to the output of a previousstage and the input of the first stage is coupled to the output of thelast stage to form a feedback circuit path. The system may also includeerror correction circuitry coupled to the feedback circuit path of thequantum oscillator and adapted to apply a deterministic error correctionto the quantum oscillator based on a difference between a measured stateof an error detection qubit in the quantum oscillator and an expectedstate of the error detection qubit.

In an embodiment, the quantum circuit for each stage is a square root ofNOT gate.

In another embodiment, the quantum circuit for each state is a Hadamardgate

In some embodiments, the quantum oscillator includes a Bell Stateoscillator (BSO), including a first stage, second stage, third stage andfourth stage. The first stage may comprise a first Bell State generator,including a first Hadamard gate and a first CNOT gate, the firstHadamard gate having an input and an output and the first CNOT gatehaving an input and an output. The second stage comprises a second BellState generator, including a second Hadamard gate and a second CNOTgate, the second Hadamard gate having an input and an output and thesecond CNOT gate having an input and an output, wherein the input of thesecond Hadamard gate is coupled to the output of the first Hadamard gateof the first Bell State generator and the input of the second CNOT gateis coupled to the output of the first CNOT gate of the first Bell Stategenerator.

A third stage of this embodiment may comprise comprising a third BellState generator, including a third Hadamard gate and a third CNOT gate,the third Hadamard gate having an input and an output and the third CNOTgate having an input and an output, wherein the input of the thirdHadamard gate is coupled to the output of the second Hadamard gate ofthe second Bell State generator and the input of the third CNOT gate iscoupled to the output of the second CNOT gate of the second Bell Stategenerator. A fourth stage comprises a fourth Bell State generator,including a fourth Hadamard gate and a fourth CNOT gate, the fourthHadamard gate having an input and an output and the fourth CNOT gatehaving an input and an output, wherein the input of the fourth Hadamardgate is coupled to the output of the third Hadamard gate of the thirdBell State generator and the input of the fourth CNOT gate is coupled tothe output of the third CNOT gate of the third Bell State generator, andwherein the feedback circuit path is formed from the coupling of theinput of the first Hadamard gate of the first Bell State generator tothe output of the fourth Hadamard gate of the fourth Bell Stategenerator and the coupling of the input of the first CNOT gate of thefirst Bell State generator to the output of the fourth CNOT gate of thefourth Bell State generator.

In another embodiment, the quantum oscillator includes a Greenberger,Home and

Zeilinger (GHZ) state oscillator including a first stage, second stage,third stage and fourth stage. The first stage comprises a first GHZstate generator, including a first Hadamard gate, a first CNOT gate anda second CNOT gate, the first Hadamard gate having an input and anoutput, the first CNOT gate having an input and an output and the secondCNOT gate having an input and an output. A second stage comprises asecond GHZ state generator, including a second Hadamard gate, a thirdCNOT gate and a fourth CNOT gate, the second Hadamard gate having aninput and an output, the third CNOT gate having an input and an output,and the fourth CNOT gate having an input and an output, wherein theinput of the second Hadamard gate is coupled to the output of the firstHadamard gate of the first GHZ state generator, the input of the thirdCNOT gate is coupled to the output of the second CNOT gate of the firstGHZ state generator and the input of the fourth CNOT gate is coupled tothe output of the second CNOT gate of the first GHZ state generator.

A third stage of the embodiment comprises a third GHZ state generator,including a third Hadamard gate, a fifth CNOT gate and a sixth CNOTgate, the third Hadamard gate having an input and an output, the fifthCNOT gate having an input and an output and the sixth CNOT gate havingan input and an output, wherein the input of the third Hadamard gate iscoupled to the output of the second Hadamard gate of the second GHZstate generator, the input of the fifth CNOT gate is coupled to theoutput of the third CNOT gate of the second GHZ state generator and theinput of the sixth CNOT gate is coupled to the output of the fourth CNOTgate of the second GHZ state generator. A fourth stage comprises afourth GHZ state generator, including a fourth Hadamard gate, a seventhCNOT gate, and an eighth CNOT gate, the fourth Hadamard gate having aninput and an output, the seventh CNOT gate having an input and anoutput, and the eighth CNOT gate having an input and an output, whereinthe input of the fourth Hadamard gate is coupled to the output of thethird Hadamard gate of the third GHZ state generator, the input of theseventh CNOT gate is coupled to the output of the fifth CNOT gate of thethird GHZ state generator, and the input of the eighth CNOT gate iscoupled to the output of the sixth CNOT gate of the third GHZ stategenerator, and wherein the feedback circuit path is formed from thecoupling of the input of the first Hadamard gate of the first GHZ stategenerator to the output of the fourth Hadamard gate of the fourth GHZstate generator, the coupling of the input of the first CNOT gate of thefirst GHZ state generator to the output of the seventh CNOT gate of thefourth GHZ state generator, and the coupling of the input of the secondCNOT gate of the first Bell State generator to the output of the eighthCNOT gate of the fourth GHZ state generator.

In a particular embodiment, a system for the quantum coherencepreservation of a qubit includes a first quantum oscillator and a secondquantum oscillator. The first quantum oscillator comprises a firstplurality of cascaded stages, each stage including a first quantumcircuit having an input and an output and adapted to evolve a qubitbetween a first state on the input and a second state on the outputwherein the stages are cascaded such that the input of one stage iscoupled to the output of a previous stage to form a first feedforwardcircuit path and the input of the first stage is coupled to the outputof the last stage to form a first feedback circuit path. The secondquantum oscillator comprises a second plurality of cascaded stages, eachstage including a second quantum circuit having an input and an outputand adapted to evolve a qubit between a first state on the input and asecond state on the output wherein the stages are cascaded such that theinput of one stage is coupled to the output of a previous stage to forma second feedforward circuit path and the input of the first stage iscoupled to the output of the last stage to form a second feedbackcircuit path. The system may also include a Fredkin gate coupling thefirst feedforward circuit path of the first quantum oscillator and thesecond feedforward circuit path of the second quantum oscillator.

In other embodiments, the system may include error correction circuitrycoupled to the first feedback circuit path of the first quantumoscillator or the second feedback circuit path of the second quantumoscillator and adapted to apply a deterministic error correction to thefirst quantum oscillator or the second quantum oscillator based on adifference between a measured state of an error detection qubit in thefirst quantum oscillator or the second quantum oscillator and anexpected state of the error detection qubit.

In one embodiment, the first quantum oscillator may be a different typeof quantum oscillator than the second type of quantum oscillator. Forexample, each of the first quantum oscillator or second quantumoscillator may be one of a quantum oscillator where the quantum circuitsof the quantum oscillator include a square root of NOT gate, a Hadamardgate, a Bell State generator or a GHZ state generator.

These, and other, aspects of the disclosure will be better appreciatedand understood when considered in conjunction with the followingdescription and the accompanying drawings. It should be understood,however, that the following description, while indicating variousembodiments of the disclosure and numerous specific details thereof, isgiven by way of illustration and not of limitation. Many substitutions,modifications, additions and/or rearrangements may be made within thescope of the disclosure without departing from the spirit thereof, andthe disclosure includes all such substitutions, modifications, additionsand/or rearrangements.

BRIEF DESCRIPTION OF THE DRAWINGS

The drawings accompanying and forming part of this specification areincluded to depict certain aspects of the disclosure. It should be notedthat the features illustrated in the drawings are not necessarily drawnto scale. A more complete understanding of the disclosure and theadvantages thereof may be acquired by referring to the followingdescription, taken in conjunction with the accompanying drawings inwhich like reference numbers indicate like features and wherein:

FIG. 1A is a block diagram of an embodiment a single qubit quantum ringoscillator circuit.

FIG. 1B is a block diagram of an embodiment of a single qubit quantumring oscillator circuit.

FIG. 2A is a block diagram of a Bell State generator.

FIG. 2B is a block diagram of a reverse Bell State generator.

FIG. 3A is a block diagram of an embodiment of a dual qubit quantum ringoscillator circuit, known as a Bell State Oscillator (BSO).

FIG. 3B is a block diagram of an embodiment of a dual qubit quantum ringoscillator circuit, known as a BSO.

FIG. 4 is a block diagram of an embodiment of a triple qubit quantumring oscillator circuit, known as a GHZ state oscillator (GSO).

FIG. 5 is a block diagram of a single qubit oscillator with errorcorrection circuitry.

FIG. 6A is a schematic block diagram of a Fredkin gate.

FIG. 6B is a block diagram of an implementation of a Fredkin gate suingtwo two-input quantum gates and one three-input quantum gate.

FIG. 6C is a block diagram of an alternative implementation of a Fredkingate using only two-input quantum gates.

DETAILED DESCRIPTION

The disclosure and the various features and advantageous details thereofare explained more fully with reference to the non-limiting embodimentsthat are illustrated in the accompanying drawings and detailed in thefollowing description. Descriptions of well-known starting materials,processing techniques, components and equipment are omitted so as not tounnecessarily obscure the invention in detail. It should be understood,however, that the detailed description and the specific examples, whileindicating some embodiments of the invention, are given by way ofillustration only and not by way of limitation. Various substitutions,modifications, additions and/or rearrangements within the spirit and/orscope of the underlying inventive concept will become apparent to thoseskilled in the art from this disclosure.

Before discussing embodiments in detail, it may be helpful to give ageneral overview of certain aspects pertaining to embodiments. As may berecalled from the above discussion, one of the main problems withquantum computing, however, is the implementation of qubits themselves.More specifically, the scale at which qubits are implemented means thatany perturbations in the qubit caused by unwanted interactions with theenvironment may result in quantum decoherence. Qubit decoherence is thusa major obstacle to the useful implementation of quantum computing.Exacerbating the problem is the fact that when several qubits are placedin close proximity to one another they can mutually interfere with eachother and, thereby, affect adjacent qubits. Sometimes this mutualinterference may be a desired effect, but when it is not, it mayintroduce considerable complexities in the quantum circuit in order totry to isolate or counteract its effects.

Accordingly, there is a need to for systems and method that can preservecoherence of one or more qubits. To that end, embodiments of thesystems, structures and quantum circuits disclosed herein achieve longerT1 or T2 times among other advantages through the continuousregeneration of a particular quantum state by repeatedly evolving one ormore qubits to the desired state and subsequently back to an eigenstateor other state. Because the subsequent states differ in a repeatablepattern, a quantum oscillator results wherein the intermediate quantumstate is the particular superimposed state that is desired to beretained. Accordingly, embodiments as disclosed herein provide quantumcircuits that repeatedly evolve oscillating qubit basis states over timefor single and multiple qubits. Embodiments of these quantum circuitsare characterized by continuously regenerating a quantum basis statethat oscillates among a subset of different basis states while alsoevolving superimposed or entangled states at other points in thecircuit.

Turning now to FIG. 1A, one embodiment of a quantum circuit for thepreservation of coherence of a single qubit is depicted. Specifically, aNOT gate or Pauli-X gate acts on a single qubit. It is the quantumequivalent of the NOT gate for classical computers (with respect to thestandard basis |0

, |1

. It equates to a rotation of the Bloch sphere around the X-axis by itradians. It maps |0

to |1

and |1

to |0

. It is represented by the Pauli matrix:

$X = {\begin{bmatrix}0 & 1 \\1 & 0\end{bmatrix}.}$A square root of NOT gate acts on a single qubit and is represented by aunitary matrix that, multiplied by itself, yields X of the NOT gate:

$\sqrt{X} = {\sqrt{NOT} = {{\frac{1}{2}\begin{bmatrix}{1 + i} & {1 - i} \\{1 - i} & {1 + i}\end{bmatrix}}.}}$

Accordingly, in one embodiment quantum coherence preservation circuit100 that is a single bit oscillator may include two square root of NOTgates 102 that are cascade with the output of the last square root ofNOT gate 120 b provided to (or fed back to) the input of the firstsquare root of NOT gate 102 a. Specifically, in these embodiments, theoutput of a first square root of NOT gate 102 a may be coupled to theinput of second square root of NOT gate 120 b. The output of the secondsquare root of NOT gate 102 b is coupled to the input of the firstsquare root of NOT gate 102 a. Thus, if a qubit input to the firstsquare root of NOT gate 102 a is in a first state, the output of thefirst square root of NOT gate 102 b is the qubit in a second stateprovided to the input of second square root of NOT gate 102 b. The qubitthat is the output of the second square root of NOT gate 102 b is in athird state, where the third state is the opposite (e.g., NOT) of thefirst state. The qubit in the third state is then fed back on the outputof the second square root of NOT gate 102 b to the input of the firstsquare root of NOT gate 102 a. After the second pass through thisquantum circuit 100, the qubit will be in the first state again. Inother words, the qubit has gone through the equivalent of two NOT gatesafter the second pass through the circuit 100. In this manner, the stateof the qubit is oscillated between the first state and its opposite(e.g., NOT the first state) and can be maintained in the quantum circuitwhile preserving the first state of the qubit maintained therein (e.g.,the state in which the qubit is initially input to the circuit).

It can be seen, then, with respect to quantum coherence preservationcircuit 100 that a qubit input to first square root of NOT gate 102 a ina basis state (e.g., |0

or |1

) will be in a superimposed state between the output of the first squareroot of NOT gate 102 a and the input of the second square root of NOTgate 102 b. After passing through the second square root of NOT gate 102b, when the qubit is output from the second square root of NOT gate 102b and fed back to the input of the first square root of NOT gate 102 ait will be in an opposite basis state (e.g., |1

or |0

). Similarly, after a second pass through the cascaded circuit, thequbit will be in the original basis state (e.g., |1

or |0

). Thus, a qubit input to the quantum ring oscillator circuit 100 in abasis state will alternate between basis states with each pass throughthe cascaded circuit 100.

Moving to FIG. 1B, another embodiment of a quantum circuit for thepreservation of coherence of a single qubit is depicted. Here, a quantumcircuit 110 that is a single qubit oscillator may include two Hadamardgates 112 that are cascaded, with the output of the last Hadamard gate112 b provided to (or fed back to) the input of the first Hadamard gate112 a. A Hadamard gate acts on a single qubit. It maps the basis state|0

to

${\frac{\left. 0 \right\rangle + \left. 1 \right\rangle}{\sqrt{2}}\mspace{14mu}{and}\mspace{14mu}\left. 1 \right\rangle\mspace{14mu}{to}\mspace{14mu}\frac{\left. 0 \right\rangle - \left. 1 \right\rangle}{\sqrt{2}}},$which means that a measurement will have equal probabilities to become 1or 0 (i.e., it creates a superposition). It represents a rotation of πabout the axis ({circumflex over (x)}+{circumflex over (z)})/√{squareroot over (2)}. Equivalently, it is the combination of two rotations, πabout the X-axis followed by

$\frac{\pi}{2}$about the Y-axis. It is represented by the Hadamard matrix:

$H = {{\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 1 \\1 & {- 1}\end{bmatrix}}.}$

Specifically, in one embodiment, the output of a first Hadamard gate 112a may be coupled to the input of a second Hadamard gate 112 b. Theoutput of the second Hadamard gate 112 b is coupled to the input of thefirst Hadamard gate 112 a. Thus, if the input qubit to the firstHadamard gate 112 a is in a first state, the output of the firstHadamard gate 112 a is a qubit in a second state provided to the inputof the second Hadamard gate 112 b, and the qubit that is output from thesecond Hadamard gate 112 b is in a third state, where the third state isequivalent to the first state. The qubit in the third state equivalentto the first state is then fed back on the output of the second Hadamardgate 112 b to the input of the first Hadamard gate. In this manner, thestate of the qubit is oscillated between the first and second states andcan be maintained in the quantum circuit while preserving the firststate of the qubit maintained therein (e.g., the state in which thequbit is initially input to the circuit).

It can be seen here with respect to circuit 110 that a qubit input tofirst Hadamard gate 112 a in an initial basis state (e.g., |0

or |1

) will be in a superimposed state between the output of the firstHadamard gate 112 a and the input of the second Hadamard gate 112 b.After passing through the second Hadamard gate 112 b, when the qubit isoutput from the second Hadamard gate 112 b and fed back to the input ofthe first Hadamard gate 112 a it will again be in the initial basisstate (e.g., |0

or |1

). Thus, a qubit input to the circuit 110 in a basis state willoscillate between an initial basis state, a superimposed state and theinitial basis state with each pass through the circuit 110.

Embodiments of quantum circuits that preserve the coherence of twoqubits will now be discussed. In particular, embodiments as describedmay utilize a quantum circuit that produces linear combinations of BellStates as output values. Various embodiments of this circuit may involvecontinuous regeneration or circulation of qubits that undergo successivesuperposition, entanglement and then decoherence operations. Theregenerative nature of this circuit and the recirculation allows thecircuit to operate in a closed-loop fashion. This permits theapplication of feedback as well as feedforward analysis and controltheory techniques for real-time improvements in operational optimizationand stability of the circuit.

As some context, two qubits that are entangled and in a state ofsuperposition are said to be in one of four different Bell States iftheir respective quantum state vector has the form:

$\left. \Phi^{+} \right\rangle = {\frac{1}{\sqrt{2}}\left( {\left. 00 \right\rangle + \left. 11 \right\rangle} \right)}$$\left. \Phi^{-} \right\rangle = {\frac{1}{\sqrt{2}}\left( {\left. 00 \right\rangle - \left. 11 \right\rangle} \right)}$$\left. \Psi^{+} \right\rangle = {\frac{1}{\sqrt{2}}\left( {\left. 01 \right\rangle + \left. 10 \right\rangle} \right)}$$\left. \Psi^{-} \right\rangle = {\frac{1}{\sqrt{2}}\left( {\left. 01 \right\rangle - \left. 10 \right\rangle} \right)}$

A quantum Bell State can be created with two elementary quantumoperations consisting of a Hadamard gate followed with a controlled-NOT(CNOT) operation. The resulting Bell State generator 200 is depicted inFIG. 2A using the notation of e.g., [DiV:98] and includes Hadamard gate210 having an input 224, the output of which is used to control CNOTgate 220 on the control input of the CNOT gate 220 with input 222 andoutput 225. If the input qubits (222, 224) are initialized to a basisstate of |0

or |1

before they are sent to the circuit input, then they are evolved into aBell State by the quantum circuit 200 in FIG. 2A.

The transfer matrix for the Bell State generator in FIG. 2A is denotedas B and is computed as follows:

$B = {{\begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 0 & 1 \\0 & 0 & 1 & 0\end{bmatrix}\left( {{\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 1 \\1 & {- 1}\end{bmatrix}} \otimes \begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}} \right)} = {\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 0 & 1 & 0 \\0 & 1 & 0 & 1 \\0 & 1 & 0 & {- 1} \\1 & 0 & {- 1} & 0\end{bmatrix}}}$

Consider the case where the qubit pair are initialized and thenrepresented as |α

and |β

. The initial quantum state can then be represented as:|α

⊕|β

=|αβ

The four Bell States that are obtained using the Bell State generatorcircuit are theoretically computed as B|αβ

when |αβ

is initialized to |00

, |01

, |10

, or |11

. As an example:

${{B\left. ❘{\alpha\beta} \right\rangle} = {{B\left. ❘00 \right\rangle} = {{{\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 0 & 1 & 0 \\0 & 1 & 0 & 1 \\0 & 1 & 0 & {- 1} \\1 & 0 & {- 1} & 0\end{bmatrix}}\begin{bmatrix}1 \\0 \\0 \\0\end{bmatrix}} = {{\frac{1}{\sqrt{2}}\begin{bmatrix}1 \\0 \\0 \\1\end{bmatrix}} = {{\frac{1}{\sqrt{2}}\left( {\left. ❘00 \right\rangle + \left. ❘11 \right\rangle} \right)} = \left. ❘\Phi^{+} \right\rangle}}}}}{{B\left. ❘{\alpha\beta} \right\rangle} = {{B\left. ❘01 \right\rangle} = {{{\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 0 & 1 & 0 \\0 & 1 & 0 & 1 \\0 & 1 & 0 & {- 1} \\1 & 0 & {- 1} & 0\end{bmatrix}}\begin{bmatrix}0 \\1 \\0 \\0\end{bmatrix}} = {{\frac{1}{\sqrt{2}}\begin{bmatrix}0 \\1 \\1 \\0\end{bmatrix}} = {{\frac{1}{\sqrt{2}}\left( {\left. ❘01 \right\rangle + \left. ❘10 \right\rangle} \right)} = \left. ❘\Psi^{+} \right\rangle}}}}}{{B\left. ❘{\alpha\beta} \right\rangle} = {{B\left. ❘10 \right\rangle} = {{{\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 0 & 1 & 0 \\0 & 1 & 0 & 1 \\0 & 1 & 0 & {- 1} \\1 & 0 & {- 1} & 0\end{bmatrix}}\begin{bmatrix}0 \\0 \\1 \\0\end{bmatrix}} = {{\frac{1}{\sqrt{2}}\begin{bmatrix}1 \\0 \\0 \\{- 1}\end{bmatrix}} = {{\frac{1}{\sqrt{2}}\left( {\left. ❘00 \right\rangle - \left. ❘11 \right\rangle} \right)} = \left. ❘\Phi^{-} \right\rangle}}}}}{{B\left. ❘{\alpha\beta} \right\rangle} = {{B\left. ❘11 \right\rangle} = {{{\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 0 & 1 & 0 \\0 & 1 & 0 & 1 \\0 & 1 & 0 & {- 1} \\1 & 0 & {- 1} & 0\end{bmatrix}}\begin{bmatrix}0 \\0 \\0 \\1\end{bmatrix}} = {{\frac{1}{\sqrt{2}}\begin{bmatrix}0 \\1 \\{- 1} \\0\end{bmatrix}} = {{\frac{1}{\sqrt{2}}\left( {\left. ❘01 \right\rangle - \left. ❘10 \right\rangle} \right)} = \left. ❘\Psi^{-} \right\rangle}}}}}$

A quantum circuit similar to the Bell State generator of FIG. 2A wherethe quantum operations are reversed in order (and whose transfer matrixis denoted as R) is depicted in FIG. 2B. Here, the qubit input toHadamard gate 230 on line 232 is used to control the operation of CNOTgate 240 on an input qubit on line 234.

Moving to FIG. 3A, a logic block diagram for one embodiment of a quantumpreservation circuit for two qubits referred to as a Bell StateOscillator (BSO) is depicted. An embodiment of a corresponding quantumcircuit for the embodiment of FIG. 3A is depicted in FIG. 3B. Here, theBSO 300 is a quantum circuit comprising a cascade or chain of fourquantum circuits 310 (e.g., 310 a, 310 b, 310 c and 310 d), each quantumcircuit 310 characterized by B (e.g., each having a transfer matrixequivalent to a Bell State generator as discussed) wherein the evolvedoutput qubit pair from the cascade is in a feedback arrangement (e.g.,the output of circuit 310 d is provided as feedback into the input ofcircuit 310 a). Such a feedback configuration is possible since thequantum state after the evolution through four consecutive B circuits310 is an eigenstate. The injection of the initial |αβ

basis state pair on input lines 302 a, 302 b may be provided as theinput to circuit 310 a and will be the basis state pair |ϕ₀

.

This embodiment of the BSO 300 may be comprised of four Bell Stategenerators 350 (e.g., 350 a, 350 b, 350 c, 350 d) with the circuit pathbetween Bell State generators 350 a and 350 d forming a feedforwardcircuit path and a feedback circuit path of feedback loop connecting theoutputs of the chain to the inputs of the chain as depicted in FIG. 3B.In other words, the outputs of one Bell State generator 350 may beprovided as the corresponding inputs to a previous Bell State generator350 in the cascade or chain. Specifically, for example, in theembodiment depicted the output of Hadamard gate 352 d of Bell Stategenerator 350 d is provided as input on line 302 a to Hadamard gate 352a of Bell State generator 350 a and the output of CNOT gate 354 d ofBell State generator 350 d is provided as input on line 302 b to CNOTgate 354 a of Bell State generator 350 a. Furthermore, the BSO 300 isinitialized by injecting a qubit pair lap) on the input lines 302 a, 302b at the quantum circuit state indicated by the dashed line denoted as|ϕ₀

. After the initialization of |ϕ₀

and the BSO evolved states |ϕ₀

, |ϕ₂

, and |ϕ₃

, the quantum state |ϕ₄

evolves to an eigenstate or basis state. The quantum states |ϕ₁

, |ϕ₂

, and |ϕ₃

are referred to as “intermediate quantum states” and the resulting basisstate |ϕ₄

as the “feedback quantum state”. Different quantum state vectorevolutions are depicted with a dashed line denoted as |ϕ₀

, |ϕ₁

, |ϕ₂

, |ϕ₃

, and |ϕ₄

.

After the initialization of |ϕ₀

(note that the quantum state |ϕ₀

=|ϕ₄

due to the feedback structure) of the depicted embodiment, theintermediate quantum states |ϕ₁

, |ϕ₂

and |ϕ₃

are entangled and superimposed qubit pairs. When |ϕ₀

=|00

, then |ϕ₄

=|01

, a basis state. Alternatively, when |ϕ₀=|01

, then |ϕ₄

=|00

, a basis state. Thus, the sequence of subsequent quantum states |ϕ₀

(or, |ϕ₄

), oscillates between |00

and |01

. However, one point of novelty of embodiments of the BSO is that theintermediate quantum states |ϕ₁

, |ϕ₂

, and |ϕ₃

are qubit pairs that are entangled and superimposed. In fact, theseintermediate states are linear combinations of Bell States.Alternatively, when |ϕ₀

=|10

, then the resulting |ϕ₄=11

, and both are also and likewise, basis states. This oscillatorybehavior is indicated through the following analysis.

Assuming|αβ

=|ϕ₀=|00

, we can analyze the evolved quantum state vectors as |ϕ₁

=B|ϕ₀

, |ϕ₂=B²|ϕ₀

, |ϕ₃

=B³|ϕ₀

, and |ϕ₄=B⁴|ϕ₀

. Thus, the oscillatory behavior is observed using the B⁴ transfermatrix.

$B^{4} = {{\left( \frac{1}{\sqrt{2}} \right)^{4}\begin{bmatrix}1 & 0 & 1 & 0 \\0 & 1 & 0 & 1 \\0 & 1 & 0 & {- 1} \\1 & 0 & {- 1} & 0\end{bmatrix}}^{4} = \begin{bmatrix}0 & 1 & 0 & 0 \\1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 \\0 & 0 & 1 & 0\end{bmatrix}}$

It is noted that B⁴ is a simple permutation matrix. Assuming that |αβ

=|ϕ₀

=|00

, the B⁴ transfer matrix may be used to illustrate the oscillatorybehavior with various initialized |αβ

=|ϕ₀

basis states.

${{B^{4}\left. ❘00 \right\rangle} = {{\begin{bmatrix}0 & 1 & 0 & 0 \\1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 \\0 & 0 & 1 & 0\end{bmatrix}\begin{bmatrix}1 \\0 \\0 \\0\end{bmatrix}} = {\begin{bmatrix}0 \\1 \\0 \\0\end{bmatrix} = \left. ❘01 \right\rangle}}}{{B^{4}\left. ❘01 \right\rangle} = {{\begin{bmatrix}0 & 1 & 0 & 0 \\1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 \\0 & 0 & 1 & 0\end{bmatrix}\begin{bmatrix}0 \\1 \\0 \\0\end{bmatrix}} = {\begin{bmatrix}1 \\0 \\0 \\0\end{bmatrix} = \left. ❘00 \right\rangle}}}{{B^{4}\left. ❘10 \right\rangle} = {{\begin{bmatrix}0 & 1 & 0 & 0 \\1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 \\0 & 0 & 1 & 0\end{bmatrix}\begin{bmatrix}0 \\0 \\1 \\0\end{bmatrix}} = {\begin{bmatrix}0 \\0 \\0 \\1\end{bmatrix} = \left. ❘11 \right\rangle}}}{{B^{4}\left. ❘11 \right\rangle} = {{\begin{bmatrix}0 & 1 & 0 & 0 \\1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 \\0 & 0 & 1 & 0\end{bmatrix}\begin{bmatrix}0 \\0 \\0 \\1\end{bmatrix}} = {\begin{bmatrix}0 \\0 \\1 \\0\end{bmatrix} = \left. ❘10 \right\rangle}}}$

One aspect of BSO 300 is that the intermediate states of the circuitlabeled as |ϕ₁

|ϕ₂

, and |ϕ₃

are comprised of qubit pairs that are entangled in various states ofsuperposition. These intermediate states are computed using B, B², andB³ transfer matrices that yield the intermediate states |ϕ₁

, |ϕ₂

, and |ϕ₃

respectively. Finally, it is noted that the intermediate states are alllinear combinations of the various Bell States, |Φ⁺

, |Φ⁻

, |ψ⁺

, and |ψ⁻

. Therefore, the BSO 300 cycles through various linear combinations ofBell States for the intermediate quantum states and a basis state in theinitialization or feedback states.

${B = {\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 0 & 1 & 0 \\0 & 1 & 0 & 1 \\0 & 1 & 0 & {- 1} \\1 & 0 & {- 1} & 0\end{bmatrix}}},{B^{2} = {\frac{1}{2}\begin{bmatrix}1 & 1 & 1 & {- 1} \\1 & 1 & {- 1} & 1 \\{- 1} & 1 & 1 & 1 \\1 & {- 1} & 1 & 1\end{bmatrix}}},{B^{3} = {\frac{1}{\sqrt{2}}\begin{bmatrix}0 & 1 & 1 & 0 \\1 & 0 & 0 & 1 \\0 & 1 & {- 1} & 0 \\1 & 0 & 0 & {- 1}\end{bmatrix}}}$

The following four examples contain the calculations that yield theintermediate quantum states |ϕ₁

, |ϕ₂

, and |ϕ₃

when the BSO is initialized with all four possible basis state pairs for|αβ

=|ϕ₀

.

EXAMPLE 1: Initialize |ϕ₀=|00:

$\left. {\left. {{\left. ❘\phi_{1} \right\rangle = {{B\left. ❘00 \right\rangle} = {{{\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 0 & 1 & 0 \\0 & 1 & 0 & 1 \\0 & 1 & 0 & {- 1} \\1 & 0 & {- 1} & 0\end{bmatrix}}\begin{bmatrix}1 \\0 \\0 \\0\end{bmatrix}} = {{\frac{1}{\sqrt{2}}\begin{bmatrix}1 \\0 \\0 \\1\end{bmatrix}} = {{\frac{1}{\sqrt{2}}\left( {\left. ❘00 \right\rangle + \left. ❘11 \right\rangle} \right)} = \left. ❘\Phi^{+} \right\rangle}}}}}{\left. ❘\phi_{2} \right\rangle = {{B^{2}\left. ❘00 \right\rangle} = {{{\frac{1}{2}\begin{bmatrix}1 & 1 & 1 & {- 1} \\1 & 1 & {- 1} & 1 \\{- 1} & 1 & 1 & 1 \\1 & {- 1} & 1 & 1\end{bmatrix}}\begin{bmatrix}1 \\0 \\0 \\0\end{bmatrix}} = {{\frac{1}{2}\begin{bmatrix}1 \\1 \\{- 1} \\1\end{bmatrix}} = {{\frac{1}{\sqrt{2}}\left( {\left. ❘00 \right\rangle + \left. ❘01 \right\rangle - \left. ❘10 \right\rangle + \left. ❘11 \right\rangle} \right)} = {\frac{1}{\sqrt{2}}\left( {\left. ❘\Phi^{+} \right\rangle + \left. ❘\Psi^{-} \right\rangle} \right)}}}}}}{\left. ❘\phi_{2} \right\rangle = {{B^{3}\left. ❘00 \right\rangle} = {{{\frac{1}{\sqrt{2}}\begin{bmatrix}0 & 1 & 1 & 0 \\1 & 0 & 0 & 1 \\0 & 1 & {- 1} & 0 \\1 & 0 & 0 & {- 1}\end{bmatrix}}\begin{bmatrix}1 \\0 \\0 \\0\end{bmatrix}} = {{\frac{1}{\sqrt{2}}\begin{bmatrix}0 \\1 \\0 \\1\end{bmatrix}} = {\frac{1}{\sqrt{2}}\left( {❘01} \right.}}}}}} \right\rangle + \left. ❘11 \right\rangle} \right) = {\frac{1}{2}\left( {\left. ❘\Phi^{+} \right\rangle - \left. ❘\Phi^{-} \right\rangle + \left. ❘\Psi^{+} \right\rangle + \left. ❘\Psi^{-} \right\rangle} \right)}$

EXAMPLE 2: Initialize |ϕ₀=|01:

${\left. ❘\phi_{1} \right\rangle = {{B\left. ❘01 \right\rangle} = {{{\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 0 & 1 & 0 \\0 & 1 & 0 & 1 \\0 & 1 & 0 & {- 1} \\1 & 0 & {- 1} & 0\end{bmatrix}}\begin{bmatrix}0 \\1 \\0 \\0\end{bmatrix}} = {{\frac{1}{\sqrt{2}}\begin{bmatrix}0 \\1 \\1 \\0\end{bmatrix}} = {{\frac{1}{\sqrt{2}}\left( {\left. ❘01 \right\rangle + \left. ❘10 \right\rangle} \right)} = \left. ❘\Psi^{+} \right\rangle}}}}}{\left. ❘\phi_{2} \right\rangle = {{B^{2}\left. ❘01 \right\rangle} = {{{\frac{1}{2}\begin{bmatrix}1 & 1 & 1 & {- 1} \\1 & 1 & {- 1} & 1 \\{- 1} & 1 & 1 & 1 \\1 & {- 1} & 1 & 1\end{bmatrix}}\begin{bmatrix}0 \\1 \\0 \\0\end{bmatrix}} = {{\frac{1}{2}\begin{bmatrix}1 \\1 \\{- 1} \\1\end{bmatrix}} = {{\frac{1}{2}\left( {\left. ❘00 \right\rangle + \left. ❘01 \right\rangle - \left. ❘10 \right\rangle + \left. ❘11 \right\rangle} \right)} = {\frac{1}{\sqrt{2}}\left( {\left. ❘\Phi^{+} \right\rangle + \left. ❘\Psi^{-} \right\rangle} \right)}}}}}}{\left. ❘\phi_{2} \right\rangle = {{B^{3}\left. ❘01 \right\rangle} = {{{\frac{1}{\sqrt{2}}\begin{bmatrix}0 & 1 & 1 & 0 \\1 & 0 & 0 & 1 \\0 & 1 & {- 1} & 0 \\1 & 0 & 0 & {- 1}\end{bmatrix}}\begin{bmatrix}0 \\1 \\0 \\0\end{bmatrix}} = {{\frac{1}{\sqrt{2}}\begin{bmatrix}1 \\0 \\1 \\0\end{bmatrix}} = {{\frac{1}{\sqrt{2}}\left( {\left. ❘00 \right\rangle + \left. ❘10 \right\rangle} \right)} = {\frac{1}{2}\left( {\left. ❘\Phi^{+} \right\rangle - \left. ❘\Phi^{-} \right\rangle + \left. ❘\Psi^{+} \right\rangle + \left. ❘\Psi^{-} \right\rangle} \right)}}}}}}$

EXAMPLE 3: Initialize |ϕ₀=|10:

$\left. {{\left. \left. {\left. {{\left. ❘\phi_{1} \right\rangle = {{B\left. ❘10 \right\rangle} = {{{\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 0 & 1 & 0 \\0 & 1 & 0 & 1 \\0 & 1 & 0 & {- 1} \\1 & 0 & {- 1} & 0\end{bmatrix}}\begin{bmatrix}0 \\0 \\1 \\0\end{bmatrix}} = {{\frac{1}{\sqrt{2}}\begin{bmatrix}1 \\0 \\0 \\{- 1}\end{bmatrix}} = {{\frac{1}{\sqrt{2}}\left( {\left. ❘00 \right\rangle - \left. ❘11 \right\rangle} \right)} = \left. ❘\Phi^{-} \right\rangle}}}}}{\left. ❘\phi_{2} \right\rangle = {{B^{2}\left. ❘10 \right\rangle} = {{{\frac{1}{2}\begin{bmatrix}1 & 1 & 1 & {- 1} \\1 & 1 & {- 1} & 1 \\{- 1} & 1 & 1 & 1 \\1 & {- 1} & 1 & 1\end{bmatrix}}\begin{bmatrix}0 \\0 \\1 \\0\end{bmatrix}} = {{\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\1 \\1\end{bmatrix}} = {\frac{1}{2}\left( {\left. ❘00 \right\rangle - \left. ❘01 \right\rangle + {❘10}} \right.}}}}}} \right\rangle + {❘11}} \right\rangle \right) = {\frac{1}{\sqrt{2}}\left( {\left. ❘\Phi^{+} \right\rangle - \left. ❘\Psi^{-} \right\rangle} \right)}}{❘\phi_{2}}} \right\rangle = {{B^{3}\left. ❘10 \right\rangle} = {{{\frac{1}{\sqrt{2}}\begin{bmatrix}0 & 1 & 1 & 0 \\1 & 0 & 0 & 1 \\0 & 1 & {- 1} & 0 \\1 & 0 & 0 & {- 1}\end{bmatrix}}\begin{bmatrix}0 \\0 \\1 \\0\end{bmatrix}} = {{\frac{1}{\sqrt{2}}\begin{bmatrix}1 \\0 \\{- 1} \\0\end{bmatrix}} = {{\frac{1}{\sqrt{2}}\left( {\left. ❘00 \right\rangle - \left. ❘10 \right\rangle} \right)} = {\frac{1}{2}\left( {\left. ❘\Phi^{+} \right\rangle - \left. ❘\Phi^{-} \right\rangle - \left. ❘\Psi^{+} \right\rangle + \left. ❘\Psi^{-} \right\rangle} \right)}}}}}$

EXAMPLE 4: Initialize |ϕ₀=|11:

${\left. ❘\phi_{1} \right\rangle = {{B\left. ❘11 \right\rangle} = {{{\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 0 & 1 & 0 \\0 & 1 & 0 & 1 \\0 & 1 & 0 & {- 1} \\1 & 0 & {- 1} & 0\end{bmatrix}}\begin{bmatrix}0 \\0 \\0 \\1\end{bmatrix}} = {{\frac{1}{\sqrt{2}}\begin{bmatrix}0 \\1 \\{- 1} \\0\end{bmatrix}} = {{\frac{1}{\sqrt{2}}\left( {\left. ❘01 \right\rangle - \left. ❘10 \right\rangle} \right)} = \left. ❘\Psi^{-} \right\rangle}}}}}{\left. ❘\phi_{2} \right\rangle = {{B^{2}\left. ❘11 \right\rangle} = {{{\frac{1}{2}\begin{bmatrix}1 & 1 & 1 & {- 1} \\1 & 1 & {- 1} & 1 \\{- 1} & 1 & 1 & 1 \\1 & {- 1} & 1 & 1\end{bmatrix}}\begin{bmatrix}0 \\0 \\0 \\1\end{bmatrix}} = {{\frac{1}{2}\begin{bmatrix}{- 1} \\1 \\1 \\1\end{bmatrix}} = {{\frac{1}{2}\left( {\left. ❘01 \right\rangle - \left. ❘00 \right\rangle + \left. ❘10 \right\rangle + \left. ❘11 \right\rangle} \right)} = {\frac{1}{\sqrt{2}}\left( {\left. ❘\Psi^{+} \right\rangle - \left. ❘\Phi^{-} \right\rangle} \right)}}}}}}{\left. ❘\phi_{2} \right\rangle = {{B^{3}\left. ❘11 \right\rangle} = {{{\frac{1}{\sqrt{2}}\begin{bmatrix}0 & 1 & 1 & 0 \\1 & 0 & 0 & 1 \\0 & 1 & {- 1} & 0 \\1 & 0 & 0 & {- 1}\end{bmatrix}}\begin{bmatrix}0 \\0 \\0 \\1\end{bmatrix}} = {{\frac{1}{\sqrt{2}}\begin{bmatrix}0 \\1 \\0 \\{- 1}\end{bmatrix}} = {{\frac{1}{\sqrt{2}}\left( {\left. ❘01 \right\rangle - \left. ❘11 \right\rangle} \right)} = {\frac{1}{2}\left( {\left. ❘\Phi^{-} \right\rangle - \left. ❘\Phi^{+} \right\rangle + \left. ❘\Psi^{+} \right\rangle + \left. ❘\Psi^{-} \right\rangle} \right)}}}}}}$

As can be seen then, embodiments of BSOs as illustrated herein exhibitoscillatory behavior. Although in this embodiment, the output of BellState generator 350 d is provided as input to Bell State generator 350 aon input lines 302 a, 302 b, other embodiments are possible. Thus, inthis embodiment quantum state |ϕ₄ is provided as feedback from theoutput of Bell State generator 350 d as the input basis state |ϕ₀ toBell State generator 350 a. However, the output of Bell State generator350 c may be provided as input to Bell State generator 350 b. Thus, inthis embodiment quantum state (e.g., |ϕ₃) would be provided as feedbackas quantum state (e.g., |ϕ₁) to Bell State generator 350 b. Theoperation of such a circuit would be somewhat different than that of theembodiment shown in FIGS. 3A and 3B, however the principal concept of aquantum/basis state feedback-based system can be considered the same forboth circuits.

As described previously, embodiments of a BSO as disclosed hereincontinually regenerate entangled EPR pairs through the recirculation ofqubit pairs in basis states. It has also been disclosed and shown hereinthat dependent upon the particular basis state of |αβ

=|ϕ₀

, different Bell States are achieved for |ϕ₁

. These were demonstrated in the Examples 1 through 4 as discussedabove. In particular, the previous analysis showed that one embodimentof a BSO has two distinct steady states based upon the qubit pairinitialization state, |ϕ₀

. When |ϕ₀=|00

or =|01

, |ϕ₁

alternatively exists in either |Φ⁺

or |ψ⁺

, both being fundamental Bell States. Likewise, when |0 ₀

=|10

or =|11

, |ϕ₁

alternatively exists in either |Φ⁻

or |ψ⁻

, that are also fundamental Bell States.

These two steady states of embodiments of a BSO are distinct anddifferent as can be observed from the overall transfer matrix structureof B⁴ (as shown above) since the first and third quadrants orsubmatrices correspond to transfer functions of a NOT gate, yielding aquantum circuit with behavior analogous to that of a conventional ringoscillator composed of an odd number of electronic digital logicinverter gates. The transfer matrix for B⁴ is reproduced below with thequadrant partitions indicated by the 2×2 all zero matrix denoted as [0]and the 2×2 transfer matrix for the single qubit operator, NOT, denotedas [N]. Thus, depending upon the initialization quantum state |ϕ₀

, embodiments of a BSO operate in accordance to the top or the bottomportion of the B⁴ transfer matrix.

$B^{4} = {\begin{bmatrix}0 & 1 & 0 & 0 \\1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 \\0 & 0 & 1 & 0\end{bmatrix} = \begin{bmatrix}N & 0 \\0 & N\end{bmatrix}}$

One of the two steady state values of |ϕ4

is either |01

or |10

depending upon the steady state of the BSO. It is noted that these two|ϕ4

basis states, each arising from one of the two different steady statesof the BSO, are simple permutations of one another. Other aspects ofembodiments such as these may be understood with reference to U.S.patent application Ser. No. 15/832,285 entitled “System and Method forQuantum Coherence Preservation of Qubits”, by Oxford et al, filed Dec.5, 2017 which is hereby incorporated by reference in its entirety.

It will be noted that other embodiments of multiple qubit quantum ringoscillator circuits may employ other quantum gates in the individualcascaded stages.

Embodiments of quantum circuits that preserve the coherence of threequbits will now be discussed. Again, various embodiments of this circuitmay involve continuous regeneration or circulation of qubits thatundergo successive superposition, entanglement and then decoherenceoperations. The regenerative nature of this circuit and therecirculation allows the circuit to operate in a closed-loop fashion.This permits the application of feedback as well as feedforward analysisand control theory techniques for real-time improvements in operationaloptimization and stability of the circuit.

Looking now at FIG. 4 , one embodiment of a quantum circuit forpreserving the coherence of three qubits is depicted. Quantumpreservation circuit 400 utilizes cascaded Greenberger, Horne andZeilinger (GHZ) state generators 402. Each GHZ state generator 402 mayinclude a Hadamard gate 404 having an input and an output and a firstCNOT gate 406 having an input and an output and controlled by the outputof the Hadamard gate 404, and a second CNOT gate 408 having an input andan output and controlled by the output of the Hadamard gate 404. Thus,quantum preservation circuit 400 that may function as a quantumoscillator for three qubits may include a cascaded set of GHZ stategenerators 402 and may be referred to as a GHZ state oscillator or GSO.

Certain embodiments of a GSO can be used to generate and preserve threeentangled qubits, and thus may be thought of as a qubit storage deviceor cell that holds the three of entangled qubits. More specifically,some embodiments of a GSO may continuously generate (or regenerate) andcirculate three qubits in a feedback loop. Such a GSO 400 may, forexample, include a set of cascaded GHZ state generators 402, with eachGHZ state generator circuit 402 providing the input to the subsequentGHZ state generator circuit 402, and the output of the final GHZ stategenerator in the chain 402 d coupled back to the input of the first GHZstate generator circuit in the chain 402 a.

Specifically, in one embodiment, GSO 400 may include a first GHZ stategenerator 402 a having a Hadamard gate 404 a with an input and anoutput, a first CNOT gate 406 a having an input and an output and acontrol coupled to the output of the Hadamard gate 404 a, and a secondfirst CNOT gate 408 a having an input and an output, and a controlcoupled to the output of the Hadamard gate 404 a. A second GHZ stategenerator 402 b is cascaded with the first GHZ state generator 402 a.The second GHZ state generator 402 b includes a Hadamard gate 404 b withan input and an output, a first CNOT gate 406 b having an input and anoutput and a control coupled to the output of the Hadamard gate 404 b,and a second CNOT gate 408 b having an input and an output, and acontrol coupled to the output of the Hadamard gate 404 b.

The second GHZ state generator 402 b is cascaded with the first GHZstate generator 402 a by coupling the output of the first CNOT gate 406a of the first GHZ state generator 402 a to the input of the first CNOTgate 406 b of the second GHZ state generator 402 b, the output of thesecond CNOT gate 408 a of the first GHZ state generator 402 a to theinput of the second CNOT gate 408 b of the second GHZ state generator402 b and the output of the Hadamard gate 404 a of the first GHZ stategenerator 402 a to the input of the Hadamard gate 404 b of the secondGHZ state generator 402 b.

A third GHZ state generator is cascaded with the second GHZ stategenerator and a fourth GHZ state generator is cascaded with the thirdGHZ state generator in a similar manner. Specifically, the third GHZstate generator 402 c is cascaded with the second GHZ state generator402 b by coupling the output of the first CNOT gate 406 b of the secondGHZ state generator 402 b to the input of the first CNOT gate 406 c ofthe third GHZ state generator 402 c, the output of the second CNOT gate408 b of the second GHZ state generator 402 b to the input of the secondCNOT gate 408 c of the third GHZ state generator 402 c and the output ofthe Hadamard gate 404 b of the second GHZ state generator 402 b to theinput of the Hadamard gate 404 c of the third GHZ state generator 402 c.

Similarly, the fourth GHZ state generator 402 d is cascaded with thethird GHZ state generator 402 c by coupling the output of the first CNOTgate 406 c of the third GHZ state generator 402 c to the input of thefirst CNOT gate 406 d of the fourth GHZ state generator 402 d, theoutput of the second CNOT gate 408 c of the third GHZ state generator402 c to the input of the second CNOT gate 408 d of the fourth GHZ stategenerator 402 d and the output of the Hadamard gate 404 b of the thirdGHZ state generator 402 c to the input of the Hadamard gate 404 d of thefourth GHZ state generator 402 d. In this manner, the circuit pathbetween GHZ state generators 402 a and 402 d forms a feedforward circuitpath of GSO 400.

A feedback path of the GSO 400 couples the output of the fourth GHZstate generator 402 d to the input of the first GHZ state generator 402a by coupling the output of the first CNOT gate 406 d of the fourth GHZstate generator 402 d to the input of the first CNOT gate 406 a of thefirst GHZ state generator 402 a, the output of the second CNOT gate 408d of the fourth GHZ state generator 402 d to the input of the secondCNOT gate 408 a of the first GHZ state generator 402 a and the output ofthe Hadamard gate 404 d of the fourth GHZ state generator 402 d to theinput of the Hadamard gate 404 a of the first GHZ state generator.

Thus, if three qubits are injected into a GSO quantum coherencepreservation circuit 400 with one qubit being on the circuit pathcoupling first CNOT gates 406 of each GHZ sate generators 402, a secondqubit on the circuit path coupling the second CNOT gates 408 of each GHZsate generators 402 and a third qubit on the circuit path coupling theHadamard gates 404 of each GHZ state generators 402.

The evolution of the qubit triplets through each individual GHZ stage ofthe GSO can be characterized as per the following:

$G = {\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 & 0 & {- 1} \\0 & 0 & 0 & 0 & 1 & 0 & {- 1} & 0 \\0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 \\0 & 1 & 0 & {- 1} & 0 & 0 & 0 & 0 \\1 & 0 & {- 1} & 0 & 0 & 0 & 0 & 0\end{bmatrix}}$

Thus, when connected in a cascaded manner (as with the BSO), thecumulative (open loop) transfer function for the GSO can be seen to be:

$G_{3}^{4} = \begin{bmatrix}0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\end{bmatrix}$

Thus, the overall transfer function of the four-stage cascade collapsesto a simple permutation matrix. Thus, if we apply the transfer functionabove to the complete series of possible 3-qubit basis state inputs, wecan observe that the closed-loop circuit will oscillate in asubstantially similar manner to the BSO circuit:

G₃⁴❘000⟩ = ❘101⟩G₃⁴❘100⟩ = ❘001⟩G₃⁴❘001⟩ = ❘100⟩G₃⁴❘101⟩ = ❘000⟩G₃⁴❘010⟩ = ❘111⟩G₃⁴❘110⟩ = ❘011⟩G₃⁴❘011⟩ = ❘110⟩G₃⁴❘111⟩ = ❘010⟩

It should be remarked that, in the notation introduced above, the “3”subscript represents the fact that the transfer function matrix shownabove is valid for a three-qubit system. As we will discuss later, themotivation for this subscript is that the same mathematical treatment offour cascaded stages of any number of similarly-connected multiple-qubitsystems (i.e., not just 2 or 3 qubits) will produce the same kind ofoscillatory behavior, regardless of the number of parallel qubit datapaths.

As described previously, embodiments of a GSO as disclosed hereincontinually regenerate maximally-entangled qubit triplets by means ofthe recirculation of these qubits though the cascaded GHZ stategenerators. Thus, the circuit described operates by cycling the qubitcarriers into and out of eigenstates. A quantum circuit with similarstructure may also be used to maximally entangle groups of four qubits.

As can be seen then, the function of maximizing qubit decoherence timemay be provided through the use of generalized quantum ring oscillatorsincluding a cascaded set of stages (such as Bell State generators, GHZgenerators, etc.), where each cascaded stage includes a Hadamard gateand a set of CNOT gates each coupled to the output of the Hadamard gate.An overall closed-circuit path for one or more qubits can be formed bythe coupling of the output of each Hadamard gate in each stage to thecorresponding Hadamard gate in the next stage in the cascaded circuit,with the output of the Hadamard gate in the last stage coupled to theinput of the Hadamard gate of the first stage in the set of cascadedstages. Feedback paths for one or more additional qubits are formed bycoupling the output of a corresponding CNOT gate in each linear stage tothe corresponding CNOT gate in the next linear stage in the cascaded setof linear stages, with the output of the CNOT gate in the last linearstage coupled to the input of the corresponding CNOT gate of the firstlinear stage in the set of cascaded set of linear stages

For quantum coherence preservation of two or more bits then, thecapability of the circuit to store an additional qubit may beaccomplished by adding another CNOT gate to each stage, where thecontrol of the CNOT gate is coupled to the Hadamard gate of the stageand the corresponding CNOT gates of each stage are coupled to form afeedback circuit path for the added qubit. Again, the feedback circuitpath is formed by the coupling of the output of each CNOT gate to thecorresponding CNOT gate in the next stage in the cascaded set ofgenerators, with the output of the CNOT gate in the last stage coupledto the input of the corresponding CNOT gate of the first stage in theset of cascaded set of stages.

As may be noticed, one advantage of embodiments as depicted herein, isthat the number of gates in such quantum ring oscillator circuits mayscale linearly with the number of qubits which such circuits canpreserve. This can be contrasted directly with other quantum circuitswhich circuitry scales exponentially (e.g., 0(N²) or 0(N³)) with thenumber of qubits. As another advantage, in certain embodiments, thequbits within such quantum ring oscillator circuits may be entangled. Assuch, they may be well suited for implementing or storing words ofqubits, as the qubits of words must usually be entangled to preformquantum computing operations utilizing such words of qubits.

Moreover, because embodiments of the quantum coherence preservationcircuits as discussed operate by oscillating the quantum qubits througha defined set of states, certain types of errors that may decreasequantum coherence time may be effectively dealt with. More specifically,in these embodiments, when a qubit is input to the quantum circuit in abasis state, the input received on the feedback loop will be a definedbasis state. Thus, for example, in a quantum circuit for coherencepreservation of single qubit using two cascaded square root of NOT gatesas depicted in FIG. 1A, if a qubit is input to the first square root ofNOT gate in a basis state (e.g., |0

or |1

) the output of the second square root of NOT gate that is fed back tothe input of the first square root of NOT gate will be in an oppositebasis state (e.g., |1

or |1

). As another example using the circuit of FIG. 1B, in a quantum circuitfor coherence preservation of a single qubit using two cascaded Hadamardgates, if a qubit is input in a basis state (e.g., |0

or |1

) the output of the second Hadamard gate that is fed back to the inputof the first square root of NOT gate will be in the same basis state(e.g., |0

or |1

).

Accordingly, by injecting one or more qubits or sequences of qubits inbasis states into such a quantum coherence preservation circuit andmeasuring the state of these qubits on the feedback path of the circuit,deterministic errors in the circuit's signal path may be measured andpotentially corrected. Specifically, by injecting one or more qubits ina known basis state (which may be referred to as error detection orancilla qubits), expected values (e.g., the same or opposite basisstate) for those error detection qubits on the feedback path may bedetermined. Any error (e.g., deviations from the expected value) inthese error detection qubits as measured on the feedback path of thecircuit are usually due to gate fidelity issues or other deterministicerrors. Based on any measured errors determined from the differencebetween the expected basis state for these error detection qubits andthe measured states of the error correction qubits, deterministic errorsin the quantum carrier signal path (such as phase shift errors or thelike) may be determined.

Error correction circuitry included in the quantum ring oscillatorcircuit may apply error correction to the circuit to correct for thesedetermined errors. For example, a deterministic phase shift may beapplied to the quantum carrier signal path to correct for these measurederrors. Other techniques for error correction in a quantum circuit areknown in the art and are fully contemplated herein.

Importantly, because of the feedback path included in embodiments of thequantum ring oscillator circuits as disclosed herein, this errorcorrection may take place without altering any of the quantuminformation (e.g., other qubits) that are being preserved or otherwisestored or run through the quantum circuit. In other, words, the errorcorrection may be applied by the error correction circuitry when thequantum circuit is in an operational state (e.g., being used for quantumcomputation) without disturbing the quantum information being used bythe quantum system or taking the quantum circuit or quantum systemoffline. Moreover, this error determination and correction may takeplace at certain regular intervals or substantially continuously as thequantum circuit is operating. Thus, as the errors in the circuit changes(e.g., as the overall circuit heats up, the quantum carrier signal pathlengthens, the operating environment changes, etc.) the differing orchanging errors may be accounted for, and the error correction adjusted.Again, such error correction may be accomplished while the quantumcircuit is in an operational state. In this manner, it is possible torun real time adjustment of quantum storage circuits to account fordeterministic errors and increase the length of time of quantumcoherence preservation.

According to some embodiments then, error measurement circuitry mayinject or place one or more qubits or patterns of error detection qubitsin a known basis state before or after a qubit which is intended to bestored or otherwise maintained in the quantum ring oscillator circuit.This qubit may be referred to as an operational or stored qubit and maybe, for example, a qubit used in the performance of quantum computingoperations by the quantum circuit or a system that includes the quantumcircuit. These error detection qubits may be injected into the feedbackloop of the quantum coherence preservation circuit such that they areinitially provided at the input of the first gate or other quantumstructure of the circuit. The error measurement circuitry of the circuitmay then determine the measured values (e.g., the same or opposite basisstate) for those error detection qubits on the feedback path. Theseerror detection qubit measurements may be used within a classicalcontrol system and a difference between the measured error detectionqubits and the expected value for those error detection qubitsdetermined. Based on this difference, methods for optimally correctingerrors in the quantum coherence preservation circuit (such as phaseshift errors or the like) may be determined. Error correction circuitryincluded in the quantum preservation circuit may apply such errorcorrection to the quantum coherence preservation circuit to correct forthese determined errors. For example, a deterministic phase shift may beapplied to the quantum coherence preservation circuit to correct forthese measured errors.

As it is only the error detection qubits that are measured, thesemeasurements and corrections can be made without decohereing orotherwise affecting any operational or stored qubits in the circuit.However, the error correction applied by the control circuitry to adjustthe quantum circuit may serve to increase the decoherence time of anysuch operational or stored qubits in the circuit.

Additionally, this error correction may take place again at a laterpoint, or effected cumulatively or substantially continuously in thesame manner. For example, such error measurement may be performedrepeatedly, or substantially continuously, utilizing the same errordetection qubits after they have passed through the quantum coherencepreservation circuit one or more additional times. Although a singlepass through the circuit may not be easily correctable (which may dependon the precision of the deterministic error correction mechanism), thecumulative error of the qubit carrier may be more easily corrected aftermultiple passes through the loop. As the initial basis state of theseerror detection qubits is known, the expected values for these errordetection qubits at any given point (e.g., after a given number ofpasses through the quantum coherence preservation circuit) may bedetermined and used to determine the errors in the quantum coherencepreservation circuit. The error correction circuitry can then applyerror correction to adjust for this newly determined error. Thus, errorcorrection can be applied and adjusted in real-time to the quantumcircuit while it is in an on-line or operational state withoutdecohereing or otherwise affecting any operational or stored qubits inthe loop, serving to greatly increase the coherence time of thoseoperational or stored qubits.

Moving now to FIG. 5 , one embodiment of a quantum coherencepreservation circuit including error correction circuitry is depicted.It will be noted here, that while this embodiment depicts a quantum ringoscillator circuit with a single feedback circuit path, error correctioncircuitry as discussed may be utilized with equal efficacy withembodiments of such circuits with multiple feedback paths for multiplequbits. Here, the quantum circuit 500 that is a single bit oscillatormay include two square root of NOT gates 502 that are cascaded, with theoutput of the last square root of NOT gate 502 b provided to (or fedback to) the input of the first square root of NOT gate 502 a. Thequantum circuit also includes error correction circuitry 550. Errorcorrection circuitry 550 includes error measurement circuitry 552adapted to inject or place one or more error detection qubits orpatterns of error detection qubits in a known basis state before orafter a stored qubit maintained in the quantum coherence preservationcircuit 500. Error correction circuitry 550 may be coupled to thefeedback path of the quantum circuit 500 such that these error detectionqubits may be injected into the feedback loop and initially provided atthe input of the first square root of NOT gate 502 a. The errormeasurement circuitry 552 of the quantum circuit 500 may then determinethe expected states (e.g., the same or opposite basis state) for thoseerror detection qubits on the feedback path after a number of circuitsof the feedback path.

Error measurement circuitry 552 may then measure the state of these oneor more error detection qubits (e.g., after a determined number of tripsaround the feedback circuit, after every trip around the feedbackcircuit, etc.) and determine a difference between the measured errordetection qubits and the expected value for those error detectionqubits. Based on this difference, error measurement circuitry 552 maydetermine an error in the quantum coherence preservation circuit 500(such as phase shift errors or the like). Error correction circuitry 500may apply error correction to the quantum coherence preservation circuitto correct or otherwise account for these determined errors. Forexample, error correction circuitry 500 may apply a deterministic phaseshift to one or more gates of the quantum coherence preservation circuit500 to correct for these measured errors.

While deterministic errors may be accounted for according to certainembodiments, it may be difficult to account for non-deterministicerrors. For example, in quantum circuits implemented using photons, onenon-deterministic error is photonic loss. Over time the chance orprobability that a photon (e.g., a photon being used as a qubit carrier)will interact with a carrier and cause the qubit to decohere increases.However, it is impossible in a quantum circuit to utilize repeaters asare typically used with in fibre channel or the like, as the use of suchrepeaters would cause the qubit to decohere. Moreover, the no-cloningtheorem states that it is impossible to create an identical copy of anarbitrary unknown quantum state.

Embodiments as disclosed herein may thus deal with the possibility ofsuch non-deterministic errors, including the issue of photonic loss, tofurther increase the coherence time of stored qubits by making acontrolled swap of states between a photon carrying a qubit and anotherphoton. By swapping the state of the qubit to a relatively newer (e.g.,a photon that has passed through a quantum circuit a fewer number oftimes) photon at certain intervals, the problem of photonic loss may bereduced, further increasing the coherence time of qubits in such quantumcircuits.

In one embodiment, for example, a Fredkin gate may be used to couple twoquantum coherence preservation circuits. A Fredkin gate is a three-qubitgate that uses a control input to determine whether the other two inputshave their respective quantum states interchanged or not.Mathematically, the transfer matrix for the Fredkin gate is expressed asthe 8×8 matrix F where the quantum state is denoted as |cxy

with |c

serving as the “control” qubit. When |c

=|1

, the superimposed state of |x

is exchanged with that of |y

and when |c

=|0

, both |x

and |y

pass through the Fredkin gate with their states of superpositionremaining unchanged. In Dirac's braket notation, F is expressed in thefollowing equation with the particular swapping cases of interestemphasized through the use of italicsF=|000

000|+|001

001|+|010

010|+|011

001|+|100

100|+|110

101|+|101|

110 |+|111

111|

In more traditional linear algebraic notation, the transfer function forF is expressed as:

$F = \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{bmatrix}$

FIGS. 6A and 6B depict, respectively, a typical notational symbol for aFredkin gate and a quantum circuit for a Fredkin gate. The Fredkin gate600 can be constructed using a 3-input Toffoli gate 610 and two CNOTgates 620 coupled as shown in FIG. 6B. The Toffoli gate 610 can beconsidered as a controlled-controlled-NOT or as a single qubit NOToperator that utilizes two additional qubits to enable its operation.The Toffoli gate 610 can be decomposed into single and two-qubitoperators by applying Barenco's decomposition theorem to the Toffoligate (see e.g., [Bar+:95]). Those operators are the single qubitHadamard gate, and the two-qubit controlled operators consisting of theCNOT and the R_(z)(π/2) rotation denoted as V.

$V = {{R_{z}\left( {\pi/2} \right)} = \begin{bmatrix}1 & 0 \\0 & i\end{bmatrix}}$

FIG. 6C thus depicts a representation of a Fredkin gate 650 as a cascadeof these types of single and dual-input (controlled qubit) gates toprovide further illustration and to indicate the quantum cost of theFredkin function. Recently, a Fredkin gate has been realizedexperimentally at the Centre for Quantum Computation & CommunicationTechnology at Griffith University in Australia (see e.g., [PHF+:16]). Inthis implementation, the quantum state is encoded on the polarization ofa photon, hence this implementation among others, may facilitateincorporation of a Fredkin gate into embodiments of a quantum coherencepreservation circuit by, for example, coupling two quantum ringoscillator circuits.

Thus, two quantum ring oscillator circuits may be coupled through theirfeedback paths, using a Fredkin gate such that the state of the photonsin each of the two quantum coherence preservation circuits may beswapped. Generally, a stored qubit may be injected into a first of thequantum ring oscillator circuits. Photonic preservation circuitry may beprovided to determine an average number of times around this quantumring oscillator structure that an individual photon can cycle withoutbeing absorbed or scattered. After a certain number of successfultransits through the structure, the photonic preservation circuitry mayelect to either release the photon from the circuit or to transfer thequantum information from one such quantum ring oscillator to a secondquantum ring oscillator circuit. The Fredkin gate coupling the twoquantum ring oscillator circuits may then be controlled (e.g., a qubitasserted on control line of the Fredkin gate) to swap the state of thephoton carrying the qubit in the first quantum ring oscillator circuitwith the state of the newly injected photon in the second quantum ringoscillator circuit. If it can be determined that the transfer issuccessful, then we can “reset” the counter that is used to measure thenumber of cycles that the photon has transited through the circuit. Inthis manner, the statistics of the state of the photon on which thequbit is stored may be effectively “refreshed”, although there may havebeen no effect on the actual physical photon, but simply on thestatistics of that particular photon.

Thus, embodiment of the quantum circuits as disclosed herein providesystems a for evolving the quantum states of one or more qubits in adeterministic and repeating fashion such that the decoherence timeinterval is maximized, thus decreasing the likelihood that decoherenceor unintentional observations occur. Additionally, such quantum circuitsmay provide the capability to measure or otherwise utilize qubits thatoscillate among basis states without disturbing the coherency of thequantum state in other portions of the structure and to provide aconvenient means for the injection and extraction of the quantuminformation carriers without disturbing or destroying the functionalityof the system or any system that incorporates such quantum circuits.

Although the invention has been described with respect to specificembodiments thereof, these embodiments are merely illustrative, and notrestrictive of the invention. The description herein of illustratedembodiments of the invention, including the description in the Summary,is not intended to be exhaustive or to limit the invention to theprecise forms disclosed herein (and in particular, the inclusion of anyparticular embodiment, feature or function within the Summary is notintended to limit the scope of the invention to such embodiment, featureor function). Rather, the description is intended to describeillustrative embodiments, features and functions in order to provide aperson of ordinary skill in the art context to understand the inventionwithout limiting the invention to any particularly described embodiment,feature or function, including any such embodiment feature or functiondescribed in the Summary. While specific embodiments of, and examplesfor, the invention are described herein for illustrative purposes only,various equivalent modifications are possible within the spirit andscope of the invention, as those skilled in the relevant art willrecognize and appreciate. As indicated, these modifications may be madeto the invention in light of the foregoing description of illustratedembodiments of the invention and are to be included within the spiritand scope of the invention. Thus, while the invention has been describedherein with reference to particular embodiments thereof, a latitude ofmodification, various changes and substitutions are intended in theforegoing disclosures, and it will be appreciated that in some instancessome features of embodiments of the invention will be employed without acorresponding use of other features without departing from the scope andspirit of the invention as set forth. Therefore, many modifications maybe made to adapt a particular situation or material to the essentialscope and spirit of the invention.

Reference throughout this specification to “one embodiment”, “anembodiment”, or “a specific embodiment” or similar terminology meansthat a particular feature, structure, or characteristic described inconnection with the embodiment is included in at least one embodimentand may not necessarily be present in all embodiments. Thus, respectiveappearances of the phrases “in one embodiment”, “in an embodiment”, or“in a specific embodiment” or similar terminology in various placesthroughout this specification are not necessarily referring to the sameembodiment. Furthermore, the particular features, structures, orcharacteristics of any particular embodiment may be combined in anysuitable manner with one or more other embodiments. It is to beunderstood that other variations and modifications of the embodimentsdescribed and illustrated herein are possible in light of the teachingsherein and are to be considered as part of the spirit and scope of theinvention.

In the description herein, numerous specific details are provided, suchas examples of components and/or methods, to provide a thoroughunderstanding of embodiments of the invention. One skilled in therelevant art will recognize, however, that an embodiment may be able tobe practiced without one or more of the specific details, or with otherapparatus, systems, assemblies, methods, components, materials, parts,and/or the like. In other instances, well-known structures, components,systems, materials, or operations are not specifically shown ordescribed in detail to avoid obscuring aspects of embodiments of theinvention. While the invention may be illustrated by using a particularembodiment, this is not and does not limit the invention to anyparticular embodiment and a person of ordinary skill in the art willrecognize that additional embodiments are readily understandable and area part of this invention.

It will also be appreciated that one or more of the elements depicted inthe drawings/figures can also be implemented in a more separated orintegrated manner, or even removed or rendered as inoperable in certaincases, as is useful in accordance with a particular application.Additionally, any signal arrows in the drawings/figures should beconsidered only as exemplary, and not limiting, unless otherwisespecifically noted.

As used herein, the terms “comprises,” “comprising,” “includes,”“including,” “has,” “having,” or any other variation thereof, areintended to cover a non-exclusive inclusion. For example, a process,product, article, or apparatus that comprises a list of elements is notnecessarily limited only those elements but may include other elementsnot expressly listed or inherent to such process, product, article, orapparatus.

Furthermore, the term “or” as used herein is generally intended to mean“and/or” unless otherwise indicated. For example, a condition A or B issatisfied by any one of the following: A is true (or present) and B isfalse (or not present), A is false (or not present) and B is true (orpresent), and both A and B are true (or present). As used herein, a termpreceded by “a” or “an” (and “the” when antecedent basis is “a” or “an”)includes both singular and plural of such term (i.e., that the reference“a” or “an” clearly indicates only the singular or only the plural).Also, as used in the description herein and throughout the claims thatfollow, the meaning of “in” includes “in” and “on” unless the contextclearly dictates otherwise.

REFERENCES

The following references will be useful to an understanding of thedisclosure and are fully incorporated herein by reference in theirentirety for all purposes.

DiVincenzo, “The Physical Implementation of Quantum Computation”,Fortschritte der Physik 48, p. 771 (2000)

Monz, T. et al. “Realization of the quantum Toffoli gate with trappedions”. Phys. Rev. Lett. 102, 040501 (2009)

[EPR:35] A. Einstein, B. Podolsky, and N. Rosen, “Can Quantum-MechanicalDescription of Physical Reality be Considered Complete?” PhysicalReview, vol. 47, pp. 777-780, May 15, 1935.

[Bell:64] J. S. Bell, “On the Einstein Podolsky Rosen Paradox,” Physics,1, 1964, pp. 195-200, 1964.

[Bell:66] J. S. Bell, “On the Problem of Hidden Variables in QuatumMechanics,” Rev. Mod. Phys., 38(3), pp. 447-452, 1966.

[FC:72] S. J. Freedman, J. F. Clauser, “Experimental Test of LocalHidden-variable Theories,” Phys. Rev. Lett., 28 (938), pp. 938-941,1972.

[Asp+:81] A. Aspect, P. Grangier, and G. Roger, “Experimental Tests ofRealistic Local Theories via Bell's Theorem,” Phys. Rev. Lett., 47 (7),pp. 460-463, 1981.

[Asp+:82] A. Aspect, J. Dalibard, and G. Roger, “Experimental Test ofBell's Inequalities Using Time-varying Analyzers,” Phys. Rev. Lett., 49(25), pp. 1804-1807, 1982.

[FTR:07] K. Fazel, M. A. Thornton, and J. E. Rice, “ESOP-based ToffoliGate Cascade Generation,” in proc. IEEE Pacific Rim Conf. onCommunications, Computers, and Signal Processing, pp. 206-209, Aug.22-24, 2007.

[NWMTD:16] P. Niemann, R. Wile, D. M. Miller, M. A. Thornton, and R.Drechsler, “QMDDs: Efficient Quantum Function Representation andManipulation,” IEEE Trans. on CAD, vol. 35, no. 1, pp. 86-99, January2016.

[MT:06] D. M. Miller and M. A. Thornton, QMDD: A Decision DiagramStructure for Reversible and Quantum Circuits, in proc. IEEE Int. Symp.on Multiple-Valued Logic, pp. 30-30 on CD-ROM, May 17-20, 2006.

[PHF+:16] R. B. Patel, J. Ho, F. Ferreyrol, T. C. Ralph, and G. J.Pryde, “A Quantum Fredkin Gate,” Science Advances, vol. 2, no. 3, Mar.4, 2016.

[Deu:85] D. Deutsch, “Quantum theory, the Church-Turing Principle andthe Universal Quantum Computer,” Proc. of Royal Society of London A,400, pp. 97-117, 1985.

[Deu:89] D. Deutsch, “Quantum Computational Networks,” Proc. of RoyalSociety of London A, 425(1868), pp. 73-90, 1989.

[DiV:98] D. P. DiVincenzo, “Quantum Gates and Circuits,” Proc. of RoyalSociety of London A, 454(1969), pp. 261-276, 1998.

[Bar+:95] A. Barenco, et al., “Elementary Gates for QuantumComputation,” quant-ph archive, March 1995.

[OBr:03] J. L. O'Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D.Branning “Demonstration of an all-optical quantum controlled NOT gate,”Nature, 426 264-267 (2003).

[OBr:07] J. L. O'Brien, “Optical Quantum Computing,” Science, 3181567-1570 (2007).

[Cer:97] N. J. Cerf, C. Adami, and P. G. Kwiat, “Optical simulation ofquantum logic,” arXiv:quant-ph/9706022v1 (1997).

[Gar:11] J. C. Garcia-Escartin and P. Chamorro-Posada, “EquivalentQuantum Circuits,” arXiv:quant-ph/1110.2998v1 (2011).

[DH:76] W. Diffie, M. Hellman, “New Directions in Crpytography,” IEEETransactions Information Theory, November, 1976.

[EI11] A. El Nagdi, K. Liu, T. P. LaFave Jr., L. R. Hunt, V.Ramakrishna, M. Dabkowski, D. L. MacFarlane, M. P. Christensen “ActiveIntegrated Filters for RF-Photonic Channelizers” Sensors 11(2) 1297-1320(2011).

[Su09] N. Sultana, W. Zhou, T. J. LaFave and D. L. MacFarlane “HBr BasedICP Etching of High Aspect Ratio Nanoscale Trenches in InP:Considerations for Photonic Applications” J. Vac. Sci. Technol. B 272351 (2009).

[Hu08] N. R. Huntoon, M. P. Christensen, D. L. MacFarlane, G. A. Evans,C. S. Yeh “Integrated Photonic Coupler Based on Frustrated TotalInternal Reflection” Appl. Opt. 47 5682 (2008).

[Zh08] W. Zhou, N. Sultana and D. L. MacFarlane “HBr-Based InductivelyCoupled Plasma Etching of High Aspect Ratio Nanoscale Trenches inGaInAsP/InP” J. Vac. Sci. Technol. B 26 1896 (2008).

[Am01] A. Ameduri, B. Boutevin, and B. Kostov “Fluoroelastomers:synthesis, properties and applications” Prog. Polym. Sci. 26 105 (2001).

[Ba03a] J. Ballato, S. Foulger, & D. W. Smith Jr. “Optical properties ofperfluorocyclobutyl polymers” J. Opt. Soc. Am. B. 20(9) 1838-1843(2003).

[Ia06] S. T. Iacono, S. M. Budy, D. Ewald, and D. W. Smith Jr. “Facilepreparation of fluorovinylene aryl ether telechelic polymers with dualfunctionality for thermal chain extension and tandem crosslinking” Chem.Commun. (46) 4844 (2006).

[Ji06] J. Jiang, C. L. Callendar, C. Blanchetiere, J. P. Noad, S. Chen,J. Ballato, and D. W. Smith Jr. “Arrayed Waveguide Gratings Based onPerfluorocyclobutane Polymers for CWDM Applications” IEEE PhotonicsTechnology Letters 18(2) 370-372 (2006).

[Ji06a] J. Jiang, C. L. Callender, C. Blanetiere, J. P. Noad, S. Chen,J. Ballato, & D. W. Smith Jr. “Property-tailorable PFCB-containingpolymers for wavelength division devices” J. Lightwave Technology 24(8)3227-3234 (2006).

[Sm02] D. W. Smith Jr, S. Chen, S. M. Kumar, J. Ballato, C. Topping, H.V. Shah and S. H. Foulger “Perfluorocyclobutyl Copolymers forMicrophotonics” Adv. Mater. 14(21) 1585 (2002).

[St99] W. H. Steier, A. Chen, S. S. Lee, S. Garner, H. Zhang, V.Chuyanov, L. R. Dalton, F. Wang, A. S. Ren, C. Zhang, G. Todorova, A.Harper, H. R. Fetterman, D. T. Chen, A. Udupa, D. Bhattachara, B. Tsap“Polymer electro-optic devices for integrated optics” Chem. Phys.245(1-3) 487-506 (1999).

[Su03] S. Suresh, R. Gulotty, Jr., S. E. Bales, M. N. Inbasekaran, M.Chartier, C. Cummins, D. W. Smith, Jr. “A novel polycarbonate for hightemperature electro-optics via azo bisphenol amines accessed by Ullmanncoupling” Polymer 44 5111 (2003).

[Su05] S. Suresh, H. Zengin, B. K. Spraul, T. Sassa, T. Wada, and D. W.Smith, Jr. “Synthesis and hyperpolarizabilities of high temperaturetriarylamine-polyene chromophores” Tetrahedron Lett. 46 3913-3916(2005).

[On17] T. Ono, R. Okamoto, M. Tanida, H. F. Hofmann, S. Takeuchi“Implementation of a quantum controlled-SWAP gate with photoniccircuits” Scientific Reports, 2017.

What is claimed is:
 1. A system for the quantum coherence preservationof a qubit, comprising: an oscillator adapted to operate in a quantummode and in a classical mode, the oscillator including a plurality ofcascaded stages, each stage including: a circuit adapted to perform acorresponding operation in either the classical mode or in the quantummode, an input, and an output, wherein the stage is adapted to evolve aqubit between a first state on the input and a second state on theoutput, wherein the stages are cascaded such that the input of one stageis coupled to the output of a previous stage and the input of a firststage of the cascaded stages is coupled to the output of a last stage ofthe cascaded stages to form a feedback circuit path, and wherein theclassical mode is adapted for performing a quantum operationsimultaneously on a plurality of independent qubits.
 2. The system ofclaim 1, wherein the circuit for each stage is a square root of NOTgate.
 3. The system of claim 1, wherein the circuit for each state is aHadamard gate.
 4. The system of claim 1, wherein the oscillator includesa Bell State oscillator (BSO), including: a first stage comprising afirst Bell State generator, including a first Hadamard gate and a firstCNOT gate, the first Hadamard gate having an input and an output and thefirst CNOT gate having an input and an output; a second stage comprisinga second Bell State generator, including a second Hadamard gate and asecond CNOT gate, the second Hadamard gate having an input and an outputand the second CNOT gate having an input and an output, wherein theinput of the second Hadamard gate is coupled to the output of the firstHadamard gate of the first Bell State generator and the input of thesecond CNOT gate is coupled to the output of the first CNOT gate of thefirst Bell State generator; a third stage comprising a third Bell Stategenerator, including a third Hadamard gate and a third CNOT gate, thethird Hadamard gate having an input and an output and the third CNOTgate having an input and an output, wherein the input of the thirdHadamard gate is coupled to the output of the second Hadamard gate ofthe second Bell State generator and the input of the third CNOT gate iscoupled to the output of the second CNOT gate of the second Bell Stategenerator; and a fourth stage comprising a fourth Bell State generator,including a fourth Hadamard gate and a fourth CNOT gate, the fourthHadamard gate having an input and an output and the fourth CNOT gatehaving an input and an output, wherein the input of the fourth Hadamardgate is coupled to the output of the third Hadamard gate of the thirdBell State generator and the input of the fourth CNOT gate is coupled tothe output of the third CNOT gate of the third Bell State generator, andwherein the feedback circuit path is formed from the coupling of theinput of the first Hadamard gate of the first Bell State generator tothe output of the fourth Hadamard gate of the fourth Bell Stategenerator and the coupling of the input of the first CNOT gate of thefirst Bell State generator to the output of the fourth CNOT gate of thefourth Bell State generator.
 5. The system of claim 1, wherein theoscillator includes a Greenberger, Horne and Zeilinger (GHZ) stateoscillator (GSO), including: a first stage comprising a first GHZ stategenerator, including a first Hadamard gate, a first CNOT gate and asecond CNOT gate, the first Hadamard gate having an input and an output,the first CNOT gate having an input and an output and the second CNOTgate having an input and an output; a second stage comprising a secondGHZ state generator, including a second Hadamard gate, a third CNOT gateand a fourth CNOT gate, the second Hadamard gate having an input and anoutput, the third CNOT gate having an input and an output, and thefourth CNOT gate having an input and an output, wherein the input of thesecond Hadamard gate is coupled to the output of the first Hadamard gateof the first GHZ state generator, the input of the third CNOT gate iscoupled to the output of the second CNOT gate of the first GHZ stategenerator and the input of the fourth CNOT gate is coupled to the outputof the second CNOT gate of the first GHZ state generator; a third stagecomprising a third GHZ state generator, including a third Hadamard gate,a fifth CNOT gate and a sixth CNOT gate, the third Hadamard gate havingan input and an output, the fifth CNOT gate having an input and anoutput and the sixth CNOT gate having an input and an output, whereinthe input of the third Hadamard gate is coupled to the output of thesecond Hadamard gate of the second GHZ state generator, the input of thefifth CNOT gate is coupled to the output of the third CNOT gate of thesecond GHZ state generator and the input of the sixth CNOT gate iscoupled to the output of the fourth CNOT gate of the second GHZ stategenerator; and a fourth stage comprising a fourth GHZ state generator,including a fourth Hadamard gate, a seventh CNOT gate, and a eighth CNOTgate, the fourth Hadamard gate having an input and an output, theseventh CNOT gate having an input and an output, and the eighth CNOTgate having an input and an output, wherein the input of the fourthHadamard gate is coupled to the output of the third Hadamard gate of thethird GHZ state generator, the input of the seventh CNOT gate is coupledto the output of the fifth CNOT gate of the third GHZ state generator,and the input of the eighth CNOT gate is coupled to the output of thesixth CNOT gate of the third GHZ state generator, and wherein thefeedback circuit path is formed from the coupling of the input of thefirst Hadamard gate of the first GHZ state generator to the output ofthe fourth Hadamard gate of the fourth GHZ state generator, the couplingof the input of the first CNOT gate of the first GHZ state generator tothe output of the seventh CNOT gate of the fourth GHZ state generator,and the coupling of the input of the second CNOT gate of the first BellState generator to the output of the eighth CNOT gate of the fourth GHZstate generator.
 6. The system of claim 1, further comprising errorcorrection circuitry adapted to apply error correction to theoscillator.
 7. The system of claim 6, wherein the error correctioncircuitry is coupled to the quantum oscillator in the feedback circuitpath of the quantum oscillator.
 8. The system of claim 6, wherein theerror correction circuitry applies error correction based on an errordetection qubit.
 9. The system of claim 8, wherein the error detectionqubit is a basis-state carrier.
 10. The system of claim 6, wherein theerror correction is non-deterministic error correction.
 11. A system forthe quantum coherence preservation of a qubit, comprising: a firstoscillator comprising a first plurality of cascaded stages such that theinput of one stage of the first plurality is coupled to the output of aprevious stage of the first plurality and the first plurality ofcascaded stages comprises a feedback circuit path, wherein each stage ofthe first plurality of cascaded stages is a quantum basis state stage oran intermediate quantum state stage and the first plurality of stagesare cascaded such the stages of the first plurality of cascaded stagesalternate between the quantum basis state stage and the intermediatequantum basis state stage; and a second oscillator comprising a secondplurality of cascaded stages such that the input of one stage of thesecond plurality is coupled to the output of a previous stage of thesecond plurality and the second plurality of cascaded stages comprises afeedback circuit path, wherein each stage of the second plurality ofcascaded stages is the quantum basis state stage or the intermediatequantum state stage and the second plurality of stages are cascaded suchthe stages of the second plurality of cascaded stages alternate betweenthe quantum basis state stage and the intermediate quantum basis statestage, wherein the first oscillator is coupled to the second oscillator,the coupling of the first oscillator to the second oscillator comprisinga coupling of a first stage of the first plurality of cascades stages ofthe first oscillator to a second stage of the second plurality ofcascade stage of the second oscillator, wherein the first stage of thefirst plurality of cascaded stages of the first oscillator is theintermediate quantum basis state stage and the second stage of thesecond plurality of cascade stage of the second oscillator is thequantum basis state stage or the intermediate quantum state stage. 12.The system of claim 11, wherein each of the first plurality of stagesincludes a square root of NOT gate or a Hadamard gate.
 13. The system ofclaim 12, wherein each of the second plurality of stages includes asquare root of NOT gate or a Hadamard gate.
 14. The system of claim 11,wherein the first oscillator or second oscillator comprises a Bell Stateoscillator (BSO).
 15. The system of claim 11, wherein the firstoscillator or second oscillator comprises a Greenberger, Horne andZeilinger (GHZ) state oscillator (GSO).
 16. The system of claim 11,wherein the first oscillator or second oscillator further comprisingerror correction circuitry adapted to apply error correction to theoscillator.
 17. The system of claim 16, wherein the error correctioncircuitry applies error correction based on an error detection qubit.18. The system of claim 17, wherein the error detection qubit is abasis-state carrier.
 19. The system of claim 16, wherein the errorcorrection is non-deterministic error correction.
 20. The system ofclaim 11, wherein the error correction circuitry is coupled to the firstoscillator or the second oscillator in the feedback circuit path of thefirst oscillator or the second oscillator.